key: cord-0030327-e2j5syfb authors: Albadrani, Mohammed; Ali, Parvez; El-Garaihy, Waleed H.; Abd El-Hafez, Hassan title: Prediction of Exchange-Correlation Energy of Graphene Sheets from Reverse Degree-Based Molecular Descriptors with Applications date: 2022-04-14 journal: Materials (Basel) DOI: 10.3390/ma15082889 sha: 2eca952a14997c784a226d601edba5adbe995e79 doc_id: 30327 cord_uid: e2j5syfb Over the past few years, the popularity of graphene as a potential 2D material has increased since graphene-based materials have applications in a variety of fields, including medicine, engineering, energy, and the environment. A large number of graphene sheets as well as an understanding of graphene’s structural hierarchy are critical to the development of graphene-based materials. For a variety of purposes, it is essential to understand the fundamental structural properties of graphene. Molecular descriptors were used in this study to investigate graphene sheets’ structural behaviour. Based on our findings, reverse degree-based molecular descriptors can significantly affect the exchange-correlation energy prediction. For the exchange-correlation energy of graphene sheets, a linear regression analysis was conducted using the reverse general inverse sum indeg descriptor, [Formula: see text]. From [Formula: see text] , a set of reverse topological descriptors can be obtained all at once as a special case, resulting in a model with a high correlation coefficient (R between 0.896 and 0.998). Used together, these reverse descriptors are graphed in relation to their response to graphene. Based on this study’s findings, it is possible to predict the exchange correlation energy as well as the geometric structures of graphene sheets with very little computational cost. Carbon is a widely studied and influential element across many scientific disciplines. Many allotropes of carbon exist, each with special properties, such as graphite, diamond, and amorphous carbon as well as fullerenes, carbon nanotubes (CNTs), and graphene [1] [2] [3] [4] [5] [6] . Graphene is at the forefront of research in fields such as physics, chemistry, and materials science, among many others. Researchers have been intrigued by graphene due to its great mechanical, transportable, optical, and thermal properties as well as its thermal stability and unique electronic structures [7] [8] [9] . Graphene is packed in a unique two-dimensional nanocarbon hexagonal lattice [10, 11] . Graphene's unique combination of characteristics strongly qualifies it for use in multiple applications, such as biosensors [9] , membranes [12] , drug delivery, tissue engineering, sensing applications [13] , photodetectors [14] , electrochemical sensors [15] , and hydrogen-based energy storage [16] . A nanostructure is composed of distinct and measurable elements, known as nanopatterns. In contrast to random patterns, these patterns follow the order of chemical and physical laws. Physical and chemical laws determine how atoms and molecules form discrete and measurable geometric structures, ranging from repeating lattices to complex shapes. Rules from the chemical graph theory can be used to analyze and predict the shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results . Φ u denotes the degree of a vertex u, and ∆ ( 2 of 19 es. Rules from the chemical graph theory can be used to analyze and predict the props of these well-defined structures [17] . In the chemical graph theory, a chemical strucis represented by a corresponding molecular graph, where vertices represent atoms edges represent bonds [18] . Molecular descriptors are commonly used in the chemical h theory to predict various properties of chemical structures. Among the many molar descriptors available, the topological molecular descriptors are a prominent 0]. Topological molecular descriptors are used to transform molecular graphs into ematical models as well as encrypt significant amounts of information about the molar structure. Topological molecular descriptors can be classified into a number of ps according to their graph parameters. Some of the well-known topological detors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based detors [24] . Researchers often prefer degree-based topological descriptors due to their licity, and some of the most popular degree-based topological descriptors are the first second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic detors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologescriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a x , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a x is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the rse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse ral inverse sum indeg descriptor by assigning specific values to the parameters . Our main objective is to offer an alternate method, with high accuracy, for computing exchange-correlation energies of graphene sheets. The DFT calculations of the exge-correlation energies of graphene sheets have the advantage of being accurate, but also have the disadvantage of being computationally expensive. Therefore, Section 2 ides a relationship between the exchange-correlation energy of graphene sheets and rse degree-based topological descriptors. Section 3 contains detailed analytical results ) is the maximum degree of the graph shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results . The reverse degree [30] of a vertex u is defined as R u = ∆ ( EER REVIEW 2 of 19 shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by RG ISI (p,q) ( Materials 2022, 15, x FOR PEER REVIEW shapes. Rules from the chemical graph theory can be used to analyze and predict the erties of these well-defined structures [17] . In the chemical graph theory, a chemical ture is represented by a corresponding molecular graph, where vertices represent and edges represent bonds [18] . Molecular descriptors are commonly used in the ch graph theory to predict various properties of chemical structures. Among the man lecular descriptors available, the topological molecular descriptors are a prom [19, 20] . Topological molecular descriptors are used to transform molecular graph mathematical models as well as encrypt significant amounts of information about th lecular structure. Topological molecular descriptors can be classified into a num groups according to their graph parameters. Some of the well-known topologic scriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-base scriptors [24] . Researchers often prefer degree-based topological descriptors due to simplicity, and some of the most popular degree-based topological descriptors are th and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithme scriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based to ical descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degre vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [3 vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first defi reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as fo ( , ) (Ǥ) = ∑ + ∈ ( ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the r general inverse sum indeg descriptor by assigning specific values to the parame and . Our main objective is to offer an alternate method, with high accuracy, for comp the exchange-correlation energies of graphene sheets. The DFT calculations of t change-correlation energies of graphene sheets have the advantage of being accura they also have the disadvantage of being computationally expensive. Therefore, Sec provides a relationship between the exchange-correlation energy of graphene shee reverse degree-based topological descriptors. Section 3 contains detailed analytical r ), as follows: RG ISI (p,q) ( 2 of 19 es from the chemical graph theory can be used to analyze and predict the propse well-defined structures [17] . In the chemical graph theory, a chemical strucesented by a corresponding molecular graph, where vertices represent atoms epresent bonds [18] . Molecular descriptors are commonly used in the chemical ry to predict various properties of chemical structures. Among the many mocriptors available, the topological molecular descriptors are a prominent ological molecular descriptors are used to transform molecular graphs into al models as well as encrypt significant amounts of information about the mocture. Topological molecular descriptors can be classified into a number of ording to their graph parameters. Some of the well-known topological declude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-4] . Researchers often prefer degree-based topological descriptors due to their and some of the most popular degree-based topological descriptors are the first Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-8] , etc. Wei et al. [29] recently introduced many reverse degree-based topologtors, inspired by their work on degree-based topological descriptors. article, molecular graphs are represented by Ǥ. denotes the degree of a nd ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a defined as = ∆(Ǥ) − + 1. ive a set of reverse degree-based topological descriptors, we first define the eral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. 1 Some reverse degree-based topological descriptors derived from the reverse erse sum indeg descriptor by assigning specific values to the parameters e reverse degree-based topological descriptors derived from the reverse general indeg descriptor. Reverse redefined third Zagreb descriptor ain objective is to offer an alternate method, with high accuracy, for computing ge-correlation energies of graphene sheets. The DFT calculations of the exrelation energies of graphene sheets have the advantage of being accurate, but ve the disadvantage of being computationally expensive. Therefore, Section 2 relationship between the exchange-correlation energy of graphene sheets and ree-based topological descriptors. Section 3 contains detailed analytical results where p and q are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters p and q. shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) (Ǥ) = ∑ + ∈ ( ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchangecorrelation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results for graphene using reverse degree topological descriptors and polynomial as well as numerical comparisons. A wide range of molecular descriptors have been proposed in the current literature, but many of them show little evidence that they correlate with any of the physical or chemical properties of the chemical structure. This section highlights the inquiry that was undertaken to determine whether reverse general inverse sum indeg descriptors possess any predictive power and whether or not they should be used in any chemical applications. In order to achieve this, we selected ten graphene sheets from one cycle to ten cycles. The molecular structures of these graphene sheets are provided in Table 2 . The exchangecorrelation energies (ECE) of these graphene sheets were obtained from the literature [31] and have been listed in Table 2 . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results The reverse general inverse sum indeg descriptors of these graphene sheets were obtained through direct calculations using the edge partition technique. For example, the reverse general inverse sum indeg descriptor for the graphene sheet C 24 (RG ISI (p,q) (C 24 )), shown in Table 2 , was obtained by the following way: The molecular graph of C 24 had 24 vertices, 30 edges, and ∆(C 24 ) = 3. Based on the reverse degrees of each of the vertices, the edges set of C 24 was partitioned into three sets: rm 11 (C 24 ), rm 12 (C 24 ), rm 22 (C 24 ) with cardinalities |rm 11 (C 24 )| = 12, |rm 12 (C 24 )| = 12, |rm 22 (C 24 )| = 6. From the definition of a reverse general inverse sum indeg descriptor, we used the following: Table 3 lists 11 reverse topological descriptors: the reverse first and second Zagreb descriptor, the reverse Randić descriptor, the reverse sum−connectivity descriptor, the reverse harmonic descriptor, the reverse hyper Zagreb descriptor, the reverse geometric−arithmetic descriptor, the reverse arithmetic−geometric descriptor, the reverse inverse sum indeg descriptor, the reverse redefined first Zagreb descriptor, and the reverse redefined third Zagreb descriptor. These descriptors were obtained by setting specific values of p and q, such as the following: (0, 1), (1, 0), −1 2 , 0 , 0, −1 2 , (0, −1), (0, 2), 1 2 , −1 , −1 2 , 1 , (1, −1), (−1, 1), (1, 1) in the reverse general inverse sum indeg descriptors (Table 1) for each graphene sheet from C 6 to C 24 . To predict the exchange-correlation energy of the graphene sheets, the following linear regression model was used: where ECE is the exchange-correlation energy of the graphene sheets from C 6 to C 32 , β is the regression model constant, α is the reverse topological descriptor coefficient, and RG ISI (p,q) is any predictor from Table 1 . This linear regression model was used in compiling Table 4 , which used SPSS software to show the regression equations of the 11 reverse topological descriptors, the correlation coefficient between the exchange-correlation energy of the graphene sheets, and the reverse topological descriptors from the data obtained from Tables 2 and 3 . Statistical quantities, such as the standard error (SE) and the F-test, were used to check the reliability of the predictive models listed in Table 4 . Based on Tables 2 and 3 , we found that the reverse topological descriptors and the exchange-correlation energy exhibit similar trends and Figures 1 and 2 illustrates this similarity. Figure 3 shows the linear relationship between the exchange-correlation energy while Figure 4 graphically depicts the predictive potential of the reverse topological descriptors via the square of the correlation R 2 with the help of the reverse topological descriptors of the studied graphene sheets using the regression model presented in Table 4 . Table 4 . Linear prediction models with statistical parameters of the exchange-correlation energy of graphene sheets from C 6 to C 32 . This section covers graphene systems, which have gained a lot of research interest across a wide range of applications due to their fascinating properties. There are numerous studies [32] [33] [34] [35] [36] [37] [38] [39] [40] dedicated to the computation of topological descriptors of graphene systems in recent years. Most of these studies are devoted to obtaining an individual formula for each topological descriptor. This article presents a general reverse degree-based topological descriptor, namely, a reverse general inverse sum indeg descriptor from which 11 other reverse degree-based topological descriptors can be obtained. To compute the general reverse inverse sum indeg descriptor for the molecular structure of the graphene under study, we considered four different cases based on the number of rows (l) and the number of benzene rings in each row (k). Initially, the case in which the number of rows and the number of rings in each row were both greater than one was considered, as shown in Figures 5 and 6 as 3D plots. For the second case, the graphene structure had only one row and more than one benzene ring. Figure 7 shows such a situation. In the third case, there was more than one row with only one benzene ring in each column, as shown in Figures 8 and 9 as 3D plots. Figure 10 represents the last case where there was only one benzene ring. Using these four cases and edge partitioning as well as degree counting and graph structure analysis, the reverse general inverse sum indeg descriptor of graphene ( Materials 2022, 15, x FOR PEER REVIEW shapes. Rules from the chemical graph theory can be used to analyze a erties of these well-defined structures [17] . In the chemical graph theo ture is represented by a corresponding molecular graph, where verti and edges represent bonds [18] . Molecular descriptors are commonly graph theory to predict various properties of chemical structures. Am lecular descriptors available, the topological molecular descripto [19, 20] . Topological molecular descriptors are used to transform mo mathematical models as well as encrypt significant amounts of inform lecular structure. Topological molecular descriptors can be classifie groups according to their graph parameters. Some of the well-kno scriptors include distance [21] , degree [22] , eccentricity [23] , and scriptors [24] . Researchers often prefer degree-based topological des simplicity, and some of the most popular degree-based topological de and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geom scriptors [28] , etc. Wei et al. [29] recently introduced many reverse de ical descriptors, inspired by their work on degree-based topological d In this article, molecular graphs are represented by Ǥ. den vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The rev vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors reverse general inverse sum indeg descriptor, denoted by where and are any rea Table 1 Some reverse degree-based topological descriptors deriv general inverse sum indeg descriptor by assigning specific values and . Using Table 1 , in Equation (1), the following 11 reverse topological descriptors for the graphene when > 1, > 1 were obtained. Using Table 1 , in Equation (2), we noted the following 11 reverse topological descriptors for the graphene when = 1, > 1 Remark 2. Using Table 1 , in Equation (2), we noted the following 11 reverse topological descriptors for the graphene when = 1, > 1 Remark 2. (Ǥ) = 32 + 58 In this case, we noted the following 11 reverse topological descriptors for the graphene as follows: In this case, we noted the following 11 reverse topological descriptors for the graphene as follows: Theorem 1. The reverse general inverse sum indeg descriptor RG ISI (p,q) ( lecular structure. Topological molecular descriptors can be classified into a numbe groups according to their graph parameters. Some of the well-known topological scriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based scriptors [24] . Researchers often prefer degree-based topological descriptors due to t simplicity, and some of the most popular degree-based topological descriptors are the f and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic scriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topo ical descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define reverse general inverse sum indeg descriptor, denoted by ( , ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reve general inverse sum indeg descriptor by assigning specific values to the parameter and . Our main objective is to offer an alternate method, with high accuracy, for compu the exchange-correlation energies of graphene sheets. The DFT calculations of the change-correlation energies of graphene sheets have the advantage of being accurate, they also have the disadvantage of being computationally expensive. Therefore, Sectio provides a relationship between the exchange-correlation energy of graphene sheets reverse degree-based topological descriptors. Section 3 contains detailed analytical res )of graphene is as follows: RG ISI (p,q) ( descriptors available, the topological molecular descriptors are a prominent . Topological molecular descriptors are used to transform molecular graphs into atical models as well as encrypt significant amounts of information about the mostructure. Topological molecular descriptors can be classified into a number of according to their graph parameters. Some of the well-known topological ders include distance [21] , degree [22] , eccentricity [23] , and spectrum-based ders [24] . Researchers often prefer degree-based topological descriptors due to their ity, and some of the most popular degree-based topological descriptors are the first ond Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic ders [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologcriptors, inspired by their work on degree-based topological descriptors. this article, molecular graphs are represented by Ǥ. denotes the degree of a , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a is defined as = ∆(Ǥ) − + 1. derive a set of reverse degree-based topological descriptors, we first define the general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. ble 1 Some reverse degree-based topological descriptors derived from the reverse l inverse sum indeg descriptor by assigning specific values to the parameters . Some reverse degree-based topological descriptors derived from the reverse general inm indeg descriptor. Reverse redefined third Zagreb descriptor r main objective is to offer an alternate method, with high accuracy, for computing hange-correlation energies of graphene sheets. The DFT calculations of the ex--correlation energies of graphene sheets have the advantage of being accurate, but so have the disadvantage of being computationally expensive. Therefore, Section 2 es a relationship between the exchange-correlation energy of graphene sheets and degree-based topological descriptors. Section 3 contains detailed analytical results and RG ISI (p,q) ( re is represented by a corresponding molecular graph, where vertices represent atoms d edges represent bonds [18] . Molecular descriptors are commonly used in the chemical aph theory to predict various properties of chemical structures. Among the many mocular descriptors available, the topological molecular descriptors are a prominent 9,20]. Topological molecular descriptors are used to transform molecular graphs into athematical models as well as encrypt significant amounts of information about the mocular structure. Topological molecular descriptors can be classified into a number of oups according to their graph parameters. Some of the well-known topological deriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based deriptors [24] . Researchers often prefer degree-based topological descriptors due to their mplicity, and some of the most popular degree-based topological descriptors are the first d second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic deriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologal descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a ertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a ertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the verse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse neral inverse sum indeg descriptor by assigning specific values to the parameters d . Our main objective is to offer an alternate method, with high accuracy, for computing e exchange-correlation energies of graphene sheets. The DFT calculations of the exange-correlation energies of graphene sheets have the advantage of being accurate, but ey also have the disadvantage of being computationally expensive. Therefore, Section 2 rovides a relationship between the exchange-correlation energy of graphene sheets and verse degree-based topological descriptors. Section 3 contains detailed analytical results Proof. The proof was built by taking the four cases into account. Case 1. From the graph structure analysis, the reverse edge partition of graphene when l > 1, k > 1 contained rm 1,1 = 3lk − 2k − l − 1 edges, rm 1,2 = 4k + 2l − 4 edges, and rm 2,2 = l + 4 edges. Then, applying the definition of the reverse general inverse sum indeg descriptor, RG ISI (p,q) ( 2 of 19 es from the chemical graph theory can be used to analyze and predict the propse well-defined structures [17] . In the chemical graph theory, a chemical strucesented by a corresponding molecular graph, where vertices represent atoms epresent bonds [18] . Molecular descriptors are commonly used in the chemical ry to predict various properties of chemical structures. Among the many mocriptors available, the topological molecular descriptors are a prominent ological molecular descriptors are used to transform molecular graphs into al models as well as encrypt significant amounts of information about the mocture. Topological molecular descriptors can be classified into a number of ording to their graph parameters. Some of the well-known topological declude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-4]. Researchers often prefer degree-based topological descriptors due to their and some of the most popular degree-based topological descriptors are the first Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-8], etc. Wei et al. [29] recently introduced many reverse degree-based topologtors, inspired by their work on degree-based topological descriptors. article, molecular graphs are represented by Ǥ. denotes the degree of a nd ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a defined as = ∆(Ǥ) − + 1. ive a set of reverse degree-based topological descriptors, we first define the eral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. 1 Some reverse degree-based topological descriptors derived from the reverse erse sum indeg descriptor by assigning specific values to the parameters e reverse degree-based topological descriptors derived from the reverse general indeg descriptor. Reverse redefined third Zagreb descriptor ain objective is to offer an alternate method, with high accuracy, for computing ge-correlation energies of graphene sheets. The DFT calculations of the exrelation energies of graphene sheets have the advantage of being accurate, but ve the disadvantage of being computationally expensive. Therefore, Section 2 relationship between the exchange-correlation energy of graphene sheets and ree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results pes. Rules from the chemical graph theory can be used to analyze and predict the propes of these well-defined structures [17] . In the chemical graph theory, a chemical struce is represented by a corresponding molecular graph, where vertices represent atoms edges represent bonds [18] . Molecular descriptors are commonly used in the chemical ph theory to predict various properties of chemical structures. Among the many molar descriptors available, the topological molecular descriptors are a prominent ,20]. Topological molecular descriptors are used to transform molecular graphs into thematical models as well as encrypt significant amounts of information about the molar structure. Topological molecular descriptors can be classified into a number of ups according to their graph parameters. Some of the well-known topological deiptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based deiptors [24] . Researchers often prefer degree-based topological descriptors due to their plicity, and some of the most popular degree-based topological descriptors are the first second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic deiptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologl descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a tex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a tex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the erse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse eral inverse sum indeg descriptor by assigning specific values to the parameters . Our main objective is to offer an alternate method, with high accuracy, for computing exchange-correlation energies of graphene sheets. The DFT calculations of the exnge-correlation energies of graphene sheets have the advantage of being accurate, but y also have the disadvantage of being computationally expensive. Therefore, Section 2 vides a relationship between the exchange-correlation energy of graphene sheets and erse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but Using Table 1 , in Equation (1), the following 11 reverse topological descriptors for the graphene when l > 1, k > 1 were obtained. Remark 1. shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . ( , ) ( , ) Corresponding Reverse Topological Descriptors (0,1) Reverse sum−connectivity descriptor (0, −1) 2 Reverse hyper Zagreb descriptor Reverse geometric−arithmetic descriptor −1 1 shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . ( , ) ( , ) Corresponding Reverse Topological Descriptors (0,1) Reverse hyper Zagreb descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . ( , ) ( , ) Corresponding Reverse Topological Descriptors (0,1) Reverse harmonic descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Table 1 . Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor. ( , ) ( , ) Corresponding Reverse Topological Descriptors (0,1) Reverse sum−connectivity descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse Randić descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse second Zagreb descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters RGA( graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results ture is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the ex- shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze an erties of these well-defined structures [17] . In the chemical graph theor ture is represented by a corresponding molecular graph, where vertic and edges represent bonds [18] . Molecular descriptors are commonly u graph theory to predict various properties of chemical structures. Am lecular descriptors available, the topological molecular descriptor [19, 20] . Topological molecular descriptors are used to transform mo mathematical models as well as encrypt significant amounts of inform lecular structure. Topological molecular descriptors can be classified groups according to their graph parameters. Some of the well-kno scriptors include distance [21] , degree [22] , eccentricity [23] , and s scriptors [24] . Researchers often prefer degree-based topological desc simplicity, and some of the most popular degree-based topological des and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geom scriptors [28] , etc. Wei et al. [29] recently introduced many reverse deg ical descriptors, inspired by their work on degree-based topological d In this article, molecular graphs are represented by Ǥ. deno vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reve vertex is defined as To derive a set of reverse degree-based topological descriptors, reverse general inverse sum indeg descriptor, denoted by where and are any rea Table 1 Some reverse degree-based topological descriptors deriv general inverse sum indeg descriptor by assigning specific values t and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . pes. Rules from the chemical graph theory can be used to analyze and predict the propes of these well-defined structures [17] . In the chemical graph theory, a chemical struce is represented by a corresponding molecular graph, where vertices represent atoms edges represent bonds [18] . Molecular descriptors are commonly used in the chemical ph theory to predict various properties of chemical structures. Among the many molar descriptors available, the topological molecular descriptors are a prominent ,20]. Topological molecular descriptors are used to transform molecular graphs into thematical models as well as encrypt significant amounts of information about the molar structure. Topological molecular descriptors can be classified into a number of ups according to their graph parameters. Some of the well-known topological deiptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based deiptors [24] . Researchers often prefer degree-based topological descriptors due to their plicity, and some of the most popular degree-based topological descriptors are the first second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic deiptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologl descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a tex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a tex is defined as To derive a set of reverse degree-based topological descriptors, we first define the erse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse eral inverse sum indeg descriptor by assigning specific values to the parameters . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Using Table 1 , in Equation (2), we noted the following 11 reverse topological descriptors for the graphene when l = 1, k > 1 (i) RG ISI (0,1) = RM 1 ( shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . RM 1 (G) = 14k + 10 (ii) RG ISI (1,0) = RM 2 ( lecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results 19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results graph theory to predict various properties of chemical structures. Among the many mo-lecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results ture is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse redefined third Zagreb descriptor Our main objective is to offer an alternate method, with high accuracy, for computing shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse arithmetic−geometric descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse sum−connectivity descriptor shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Table 1 . Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor. Case 3. For l > 1, k = 1, the reverse edge partition of the graphene contains rm 1,1 = 2l − 3 edges, rm 1,2 = 2l edges, and rm 2,2 = l + 4 edges. Using the definition of the reverse general inverse sum indeg descriptor, RG ISI (p,q) ( Materials 2022, 15, x FOR PEER REVIEW shapes. Rules from the chemical graph theory can be used to analyze an erties of these well-defined structures [17] . In the chemical graph theor ture is represented by a corresponding molecular graph, where vertic and edges represent bonds [18] . Molecular descriptors are commonly u graph theory to predict various properties of chemical structures. Am lecular descriptors available, the topological molecular descriptor [19, 20] . Topological molecular descriptors are used to transform mo mathematical models as well as encrypt significant amounts of inform lecular structure. Topological molecular descriptors can be classified groups according to their graph parameters. Some of the well-kno scriptors include distance [21] , degree [22] , eccentricity [23] , and s scriptors [24] . Researchers often prefer degree-based topological desc simplicity, and some of the most popular degree-based topological des and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geom scriptors [28] , etc. Wei et al. [29] recently introduced many reverse deg ical descriptors, inspired by their work on degree-based topological d In this article, molecular graphs are represented by Ǥ. deno vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reve vertex is defined as To derive a set of reverse degree-based topological descriptors, reverse general inverse sum indeg descriptor, denoted by where and are any rea Table 1 Some reverse degree-based topological descriptors deriv general inverse sum indeg descriptor by assigning specific values t ), shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results lecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results Using Table 1 , in Equation (3), we noted the following reverse topological descriptors for the graphene when l > 1, k = 1 Remark 3. (i) RG ISI (0,1) = RM 1 ( VIEW 2 of 19 shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ) = 10 3 l + 5 2 (x) RG ISI (−1,1) = RReZG 1 ( lecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) (Ǥ) = ∑ + ∈ ( ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Table 1 . Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor. r main objective is to offer an alternate method, with high accuracy, for computing ange-correlation energies of graphene sheets. The DFT calculations of the exorrelation energies of graphene sheets have the advantage of being accurate, but have the disadvantage of being computationally expensive. Therefore, Section 2 a relationship between the exchange-correlation energy of graphene sheets and egree-based topological descriptors. Section 3 contains detailed analytical results ) = ∑ uv∈E 1 ( shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Table 1 . Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor. Data Availability Statement: All the raw data supporting the conclusion of this paper were provided by the authors. Helical microtubules of graphitic carbon Electric field effect in atomically thin carbon films Review graphite Amorphous carbon Vapor growth of diamond on diamond and other surfaces Cytotoxicity of three graphene-related materials in rainbow trout primary hepatocytes is not associated to cellular internalization Effect of graphene material structure and iron oxides deposition method on morphology and properties of graphene/iron oxide hybrids Graphene and graphene oxide for bio-sensing: General properties and the effects of graphene ripples Behavior of graphene under glow discharge plasma Fabrication of fanlike L-shaped graphene nanostructures with enhanced thermal/electrochemical properties via laser irradiation Ceramic-supported graphene oxide membrane bioreactor for the anaerobic decolorization of azo dyes Graphene nanoplatelets can improve the performances of graphene oxide-Polyaniline composite gas sensing aerogels Graphene photodetectors with asymmetric device structures on silicon chips Highly sensitive and disposable screen-printed ionic liquid/graphene based electrochemical sensors Metal hydride-Graphene composites for hydrogen based energy storage The graph description of chemical structures Chemical graphs Handbook of Molecular Descriptors Molecular descriptors. In Handbook of Computational Chemistry Correlation of heats of isomerization, and differences in heats of vaporization of isomers, among the paraffin hydrocarbons Milovanović, I. Beyond the Zagreb indices Connective eccentricity index: A novel topological descriptor for predicting biological activity Predictive potential of eigenvalue-based topological molecular descriptors Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons Characterization of molecular branching On general sum-connectivity index Bond additive modeling 2. Mathematical properties of max-min rodeg index On topological indices of remdesivir compound used in treatment of Corona virus (COVID 19) On the sum connectivity reverse index of oxide and honeycomb networks Relation Between Topological Indices and Exchange-Correlation Energy for Graphene Nanosurface: A DFT Study. Fuller. Nanotub. Carbon Nanostructures Computation of Topological Indices of Graphene Computation of New Degree-Based Topological Indices of Graphene Some results on topological indices of graphene Computation of certain topological coindices of graphene sheet and () nanotubes and nanotorus Topological properties of Graphene using some novel neighborhood degree-based topological indices Some Topological Indices and Their Polynomials of Graphene Vertex-Degree Based Topological Indices of Graphene Some edge degree based topological indices of Graphene Graph entropies of porous graphene using topological indices Acknowledgments: Authors would like to thank the Deanship of Scientific Research, Qassim University, for funding publication of this project. The authors declare that they have no competing interest. RG ISI (p,q) ( lar structure. Topological molecular descriptors can be classified into a number of ups according to their graph parameters. Some of the well-known topological deiptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based deiptors [24] . Researchers often prefer degree-based topological descriptors due to their plicity, and some of the most popular degree-based topological descriptors are the first second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic deiptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologl descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a tex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a tex is defined as = ∆(Ǥ) − + 1.To derive a set of reverse degree-based topological descriptors, we first define the erse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) (Ǥ) = ∑ + ∈ ( ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse eral inverse sum indeg descriptor by assigning specific values to the parameters . le 1. Some reverse degree-based topological descriptors derived from the reverse general inse sum indeg descriptor.( , ) ( , ) Corresponding Reverse Topological Descriptors (0,1) ( Our main objective is to offer an alternate method, with high accuracy, for computing exchange-correlation energies of graphene sheets. The DFT calculations of the exnge-correlation energies of graphene sheets have the advantage of being accurate, but y also have the disadvantage of being computationally expensive. Therefore, Section 2 vides a relationship between the exchange-correlation energy of graphene sheets and erse degree-based topological descriptors. Section 3 contains detailed analytical results ) = ∑ uv∈E 1 ( mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1.To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows: ( , ) (Ǥ) = ∑ + ∈ ( ) where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical resultsmathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1.To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1. To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical resultsRReZG 1 ( [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined asTo derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical resultslecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1.To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical resultsRReZG 3 ( and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined as = ∆(Ǥ) − + 1.To derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results ) = 32l + 58 Case 4. When l = 1, k = 1, the reverse edge partition of the graphene contained only rm 1,1 = 6 edges, and by the definition of reverse general inverse sum indeg descriptor, RG ISI (p,q) ( shapes. Rules from the chemical graph theory can be used to analyze and pred erties of these well-defined structures [17] . In the chemical graph theory, a che ture is represented by a corresponding molecular graph, where vertices repr and edges represent bonds [18] . Molecular descriptors are commonly used in t graph theory to predict various properties of chemical structures. Among th lecular descriptors available, the topological molecular descriptors are a [19, 20] . Topological molecular descriptors are used to transform molecular mathematical models as well as encrypt significant amounts of information ab lecular structure. Topological molecular descriptors can be classified into a groups according to their graph parameters. Some of the well-known top scriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum scriptors [24] . Researchers often prefer degree-based topological descriptors simplicity, and some of the most popular degree-based topological descriptors and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−ar scriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-bas ical descriptors, inspired by their work on degree-based topological descriptoIn this article, molecular graphs are represented by Ǥ. denotes the vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse deg vertex is defined as = ∆(Ǥ) − + 1.To derive a set of reverse degree-based topological descriptors, we firs reverse general inverse sum indeg descriptor, denoted by ( , ) where and are any real numbe Table 1 Some reverse degree-based topological descriptors derived from general inverse sum indeg descriptor by assigning specific values to the pa and . Our main objective is to offer an alternate method, with high accuracy, for the exchange-correlation energies of graphene sheets. The DFT calculation change-correlation energies of graphene sheets have the advantage of being a they also have the disadvantage of being computationally expensive. Therefo provides a relationship between the exchange-correlation energy of graphene reverse degree-based topological descriptors. Section 3 contains detailed analy ), RG ISI (p,q) ( ules from the chemical graph theory can be used to analyze and predict the propthese well-defined structures [17] . In the chemical graph theory, a chemical strucpresented by a corresponding molecular graph, where vertices represent atoms es represent bonds [18] . Molecular descriptors are commonly used in the chemical eory to predict various properties of chemical structures. Among the many modescriptors available, the topological molecular descriptors are a prominent Topological molecular descriptors are used to transform molecular graphs into atical models as well as encrypt significant amounts of information about the motructure. Topological molecular descriptors can be classified into a number of according to their graph parameters. Some of the well-known topological deinclude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de- [24] . Researchers often prefer degree-based topological descriptors due to their ty, and some of the most popular degree-based topological descriptors are the first nd Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de- [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologriptors, inspired by their work on degree-based topological descriptors. his article, molecular graphs are represented by Ǥ.denotes the degree of a , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a is defined as = ∆(Ǥ) − + 1. derive a set of reverse degree-based topological descriptors, we first define the general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. le 1 Some reverse degree-based topological descriptors derived from the reverse inverse sum indeg descriptor by assigning specific values to the parameters ome reverse degree-based topological descriptors derived from the reverse general inindeg descriptor. Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical resultsIn this case, we noted the following 11 reverse topological descriptors for the graphene as follows: (i) RM 1 ( 2 of 19 les from the chemical graph theory can be used to analyze and predict the propese well-defined structures [17] . In the chemical graph theory, a chemical strucresented by a corresponding molecular graph, where vertices represent atoms represent bonds [18] . Molecular descriptors are commonly used in the chemical ory to predict various properties of chemical structures. Among the many moscriptors available, the topological molecular descriptors are a prominent pological molecular descriptors are used to transform molecular graphs into ical models as well as encrypt significant amounts of information about the moucture. Topological molecular descriptors can be classified into a number of cording to their graph parameters. Some of the well-known topological deinclude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-24]. Researchers often prefer degree-based topological descriptors due to their , and some of the most popular degree-based topological descriptors are the first d Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-28], etc. Wei et al. [29] recently introduced many reverse degree-based topologptors, inspired by their work on degree-based topological descriptors. is article, molecular graphs are represented by Ǥ.denotes the degree of a and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a s defined as = ∆(Ǥ) − + 1.rive a set of reverse degree-based topological descriptors, we first define the eneral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. 1 Some reverse degree-based topological descriptors derived from the reverse verse sum indeg descriptor by assigning specific values to the parameters me reverse degree-based topological descriptors derived from the reverse general inndeg descriptor. Reverse redefined third Zagreb descriptor ain objective is to offer an alternate method, with high accuracy, for computing nge-correlation energies of graphene sheets. The DFT calculations of the exrrelation energies of graphene sheets have the advantage of being accurate, but ave the disadvantage of being computationally expensive. Therefore, Section 2 relationship between the exchange-correlation energy of graphene sheets and gree-based topological descriptors. Section 3 contains detailed analytical results ) = (6) shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined asTo derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results ( 2 of 19 es from the chemical graph theory can be used to analyze and predict the propse well-defined structures [17] . In the chemical graph theory, a chemical strucsented by a corresponding molecular graph, where vertices represent atoms epresent bonds [18] . Molecular descriptors are commonly used in the chemical y to predict various properties of chemical structures. Among the many mocriptors available, the topological molecular descriptors are a prominent ological molecular descriptors are used to transform molecular graphs into al models as well as encrypt significant amounts of information about the mocture. Topological molecular descriptors can be classified into a number of rding to their graph parameters. Some of the well-known topological declude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-] . Researchers often prefer degree-based topological descriptors due to their nd some of the most popular degree-based topological descriptors are the first Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-], etc. Wei et al. [29] recently introduced many reverse degree-based topologtors, inspired by their work on degree-based topological descriptors. article, molecular graphs are represented by Ǥ.denotes the degree of a d ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a defined asive a set of reverse degree-based topological descriptors, we first define the eral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Some reverse degree-based topological descriptors derived from the reverse erse sum indeg descriptor by assigning specific values to the parameters e reverse degree-based topological descriptors derived from the reverse general indeg descriptor. Reverse redefined third Zagreb descriptor in objective is to offer an alternate method, with high accuracy, for computing e-correlation energies of graphene sheets. The DFT calculations of the exelation energies of graphene sheets have the advantage of being accurate, but ve the disadvantage of being computationally expensive. Therefore, Section 2 elationship between the exchange-correlation energy of graphene sheets and ree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors. In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined asTo derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results ( 2 of 19 es from the chemical graph theory can be used to analyze and predict the propse well-defined structures [17] . In the chemical graph theory, a chemical strucesented by a corresponding molecular graph, where vertices represent atoms epresent bonds [18] . Molecular descriptors are commonly used in the chemical ry to predict various properties of chemical structures. Among the many mocriptors available, the topological molecular descriptors are a prominent ological molecular descriptors are used to transform molecular graphs into al models as well as encrypt significant amounts of information about the mocture. Topological molecular descriptors can be classified into a number of ording to their graph parameters. Some of the well-known topological declude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-4]. Researchers often prefer degree-based topological descriptors due to their nd some of the most popular degree-based topological descriptors are the first Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-8], etc. Wei et al. [29] recently introduced many reverse degree-based topologtors, inspired by their work on degree-based topological descriptors. article, molecular graphs are represented by Ǥ. denotes the degree of a d ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a defined as = ∆(Ǥ) − + 1.ive a set of reverse degree-based topological descriptors, we first define the eral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Some reverse degree-based topological descriptors derived from the reverse erse sum indeg descriptor by assigning specific values to the parameters e reverse degree-based topological descriptors derived from the reverse general indeg descriptor. Reverse redefined third Zagreb descriptor ain objective is to offer an alternate method, with high accuracy, for computing ge-correlation energies of graphene sheets. The DFT calculations of the exrelation energies of graphene sheets have the advantage of being accurate, but ve the disadvantage of being computationally expensive. Therefore, Section 2 relationship between the exchange-correlation energy of graphene sheets and ree-based topological descriptors. Section 3 contains detailed analytical results shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined asTo derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse inverse sum indeg descriptor (−1,1)Reverse redefined first Zagreb descriptor (1,1) Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and reverse degree-based topological descriptors. Section 3 contains detailed analytical results ( 2 of 19 les from the chemical graph theory can be used to analyze and predict the propese well-defined structures [17] . In the chemical graph theory, a chemical strucresented by a corresponding molecular graph, where vertices represent atoms represent bonds [18] . Molecular descriptors are commonly used in the chemical ory to predict various properties of chemical structures. Among the many moscriptors available, the topological molecular descriptors are a prominent pological molecular descriptors are used to transform molecular graphs into ical models as well as encrypt significant amounts of information about the moucture. Topological molecular descriptors can be classified into a number of cording to their graph parameters. Some of the well-known topological deinclude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-24] . Researchers often prefer degree-based topological descriptors due to their , and some of the most popular degree-based topological descriptors are the first d Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-28], etc. Wei et al. [29] recently introduced many reverse degree-based topologptors, inspired by their work on degree-based topological descriptors. is article, molecular graphs are represented by Ǥ.denotes the degree of a and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a is defined aserive a set of reverse degree-based topological descriptors, we first define the eneral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. 1 Some reverse degree-based topological descriptors derived from the reverse verse sum indeg descriptor by assigning specific values to the parameters me reverse degree-based topological descriptors derived from the reverse general inindeg descriptor. Reverse redefined third Zagreb descriptor ain objective is to offer an alternate method, with high accuracy, for computing nge-correlation energies of graphene sheets. The DFT calculations of the exrrelation energies of graphene sheets have the advantage of being accurate, but have the disadvantage of being computationally expensive. Therefore, Section 2 a relationship between the exchange-correlation energy of graphene sheets and ) = 2 (6) shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined asTo derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2 provides a relationship between the exchange-correlation energy of graphene sheets and ( 2 of 19 les from the chemical graph theory can be used to analyze and predict the propese well-defined structures [17] . In the chemical graph theory, a chemical strucresented by a corresponding molecular graph, where vertices represent atoms represent bonds [18] . Molecular descriptors are commonly used in the chemical ory to predict various properties of chemical structures. Among the many moscriptors available, the topological molecular descriptors are a prominent pological molecular descriptors are used to transform molecular graphs into ical models as well as encrypt significant amounts of information about the moucture. Topological molecular descriptors can be classified into a number of cording to their graph parameters. Some of the well-known topological deinclude distance [21] , degree [22] , eccentricity [23] , and spectrum-based de-24] . Researchers often prefer degree-based topological descriptors due to their , and some of the most popular degree-based topological descriptors are the first d Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic de-28], etc. Wei et al. [29] recently introduced many reverse degree-based topologptors, inspired by their work on degree-based topological descriptors. is article, molecular graphs are represented by Ǥ.denotes the degree of a and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a is defined asrive a set of reverse degree-based topological descriptors, we first define the eneral inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. 1 Some reverse degree-based topological descriptors derived from the reverse verse sum indeg descriptor by assigning specific values to the parameters me reverse degree-based topological descriptors derived from the reverse general inindeg descriptor. Reverse redefined third Zagreb descriptor ain objective is to offer an alternate method, with high accuracy, for computing nge-correlation energies of graphene sheets. The DFT calculations of the exrrelation energies of graphene sheets have the advantage of being accurate, but ave the disadvantage of being computationally expensive. Therefore, Section 2 shapes. Rules from the chemical graph theory can be used to analyze and predict the properties of these well-defined structures [17] . In the chemical graph theory, a chemical structure is represented by a corresponding molecular graph, where vertices represent atoms and edges represent bonds [18] . Molecular descriptors are commonly used in the chemical graph theory to predict various properties of chemical structures. Among the many molecular descriptors available, the topological molecular descriptors are a prominent [19, 20] . Topological molecular descriptors are used to transform molecular graphs into mathematical models as well as encrypt significant amounts of information about the molecular structure. Topological molecular descriptors can be classified into a number of groups according to their graph parameters. Some of the well-known topological descriptors include distance [21] , degree [22] , eccentricity [23] , and spectrum-based descriptors [24] . Researchers often prefer degree-based topological descriptors due to their simplicity, and some of the most popular degree-based topological descriptors are the first and second Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic descriptors [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topological descriptors, inspired by their work on degree-based topological descriptors.In this article, molecular graphs are represented by Ǥ. denotes the degree of a vertex , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a vertex is defined asTo derive a set of reverse degree-based topological descriptors, we first define the reverse general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:where and are any real numbers. Table 1 Some reverse degree-based topological descriptors derived from the reverse general inverse sum indeg descriptor by assigning specific values to the parameters and . Reverse inverse sum indeg descriptor (−1,1)Reverse redefined first Zagreb descriptor (1,1) Our main objective is to offer an alternate method, with high accuracy, for computing the exchange-correlation energies of graphene sheets. The DFT calculations of the exchange-correlation energies of graphene sheets have the advantage of being accurate, but they also have the disadvantage of being computationally expensive. Therefore, Section 2. Rules from the chemical graph theory can be used to analyze and predict the propf these well-defined structures [17] . In the chemical graph theory, a chemical strucrepresented by a corresponding molecular graph, where vertices represent atoms ges represent bonds [18] . Molecular descriptors are commonly used in the chemical theory to predict various properties of chemical structures. Among the many modescriptors available, the topological molecular descriptors are a prominent . Topological molecular descriptors are used to transform molecular graphs into matical models as well as encrypt significant amounts of information about the mostructure. Topological molecular descriptors can be classified into a number of according to their graph parameters. Some of the well-known topological ders include distance [21] , degree [22] , eccentricity [23] , and spectrum-based ders [24] . Researchers often prefer degree-based topological descriptors due to their city, and some of the most popular degree-based topological descriptors are the first cond Zagreb [25] , Randić [26] , sum−connectivity [27] , and geometric−arithmetic ders [28] , etc. Wei et al. [29] recently introduced many reverse degree-based topologscriptors, inspired by their work on degree-based topological descriptors. this article, molecular graphs are represented by Ǥ. denotes the degree of a , and ∆(Ǥ) is the maximum degree of the graph Ǥ. The reverse degree [30] of a is defined as = ∆(Ǥ) − + 1.o derive a set of reverse degree-based topological descriptors, we first define the e general inverse sum indeg descriptor, denoted by ( , ) (Ǥ) , as follows:, ) (Ǥ) = ∑ + ∈ ( ) where and are any real numbers. able 1 Some reverse degree-based topological descriptors derived from the reverse l inverse sum indeg descriptor by assigning specific values to the parameters . Some reverse degree-based topological descriptors derived from the reverse general inm indeg descriptor. In Table 5 , the numerical values of the 11 reverse topological descriptors calculated with graphene's analytical expressions when l > 1, k > 1 are presented. From Table 5 , it is possible to see how individual reverse topological descriptor differ and how they are similar. The computational results show that reverse topological descriptors are highly dependent on the values of l and k. As these values increase, the magnitude of all reverse descriptors also increases, and this can be visualized by the 3D graphical representation in Figure 11 . Figure 11 . An interactive visualization of Table 5 . In this paper, we presented a reverse general inverse sum inverse degree descriptor ( , ) from which one can derive a set of reverse degree-based topological descriptors. In order to assess the predictive potential of ( , ) , we selected the exchange-correlation energy of the graphene sheets as a data example. Based on the results obtained in this article, we can summarize them as follows:•The regression models (Table 4 ) derived from reverse topological descriptors in the present article were extremely accurate for predicting the exchange-correlation energies in the graphene sheets. Figure 11 . An interactive visualization of Table 5 . In this paper, we presented a reverse general inverse sum inverse degree descriptor RG ISI (p,q) from which one can derive a set of reverse degree-based topological descriptors. In order to assess the predictive potential of RG ISI (p,q) , we selected the exchange-correlation energy of the graphene sheets as a data example. Based on the results obtained in this article, we can summarize them as follows: The regression models (Table 4 ) derived from reverse topological descriptors in the present article were extremely accurate for predicting the exchange-correlation energies in the graphene sheets. The reverse sum−connectivity descriptor with R 2 = 0.997 was the best predictor among the 11 studied descriptors. Meanwhile, the reverse redefined first Zagreb descriptor performed poorly. The density functional theory (DFT) calculations of the electronic structure, such as the exchange-correlation energies of the graphene sheets, were precise; however, they were computationally expensive while the reverse topological descriptors models presented herein required minimal computations and provided high levels of accuracy. • Analytical expressions of the reverse first and second Zagreb descriptor, reverse Randić descriptor, reverse sum−connectivity descriptor, reverse harmonic descriptor, reverse hyper Zagreb descriptor, reverse geometric−arithmetic descriptor, reverse inverse sum indeg descriptor, and reverse redefined first and third Zagreb descriptors have been obtained for graphene structures.