key: cord-0937799-lusaqwdg authors: Rajan, R. Sundara; Kumar, K. Jagadeesh; Shantrinal, A. Arul; Rajalaxmi, T. M.; Rajasingh, Indra; Balasubramanian, Krishnan title: Biochemical and phylogenetic networks-I: hypertrees and corona products date: 2021-02-06 journal: J Math Chem DOI: 10.1007/s10910-020-01194-3 sha: eb079ccc48ed40bbcf486eda998cc9d9901dfb14 doc_id: 937799 cord_uid: lusaqwdg We have obtained graph-theoretically based topological indices for the characterization of certain graph theoretical networks of biochemical interest. We have derived certain distance, degree and eccentricity based topological indices for various k-level hypertrees and corona product of hypertrees. We have also pointed out errors in a previous study. The validity of our results is supported by computer codes for the respective indices. Several biochemical applications are pointed out. Mathematical Modeling is a predominant decision tool that can be efficacious to evaluate the transmission of a disease and apply the necessary control measures [1] [2] [3] . Analysis of changes in transmission rates helps to identify the effectiveness of the control measures and promote alternate interventions [4] . Phenomenological models were presented in [5] , to predict the dynamics of COVID-19. Artificial intelligence (AI) techniques play an important role in finding high-quality prognostic models for the analysis of infectious diseases [6] . Through Social Internet of Things (SIoT) Wang et al. [7] obtained risk-awareness for suspected COVID-19 cases using a graph embedding technique. Graph theory deals with the mathematical study and analysis of networks. These networks play a vital role in the environment and public health, and as a result ecological and epidemiological researchers have now drawn their attention to network analysis [8] . In biomedical research, graphs can capture the underlying connectivity relations among biological entities such as genes, DNA and proteins [9] [10] [11] [12] . Topological analysis of large-scale protein interaction networks can provide insights into redundancies which can in turn result in predictions of protein functions [13] . Moreover, the control strategies for infectious diseases often relies on graph theoretical networks [14] . It has been shown that the early diagnosis of neurological disorders can be made possible through the detection of abnormal patterns of neural synchronization in specific brain regions [15] . Trees are connected graphs that do not contain a cycle. The vertices of a rooted tree could represent a variety of biological entities such as DNA sequences or various species and the edges could represent their mutations or inter connections with other species. Such rooted trees of biological interest are called phylogenetic trees or evolutionary trees. A hypertree is an interconnection topology which is a combination of the binary tree and hypercube concept. Hypertrees are applicable in a scenario where there are interactions among certain vertices at a given level of the binary trees, and hence one can model such interactions through the use of hypertrees. Machine learning and AI techniques have shown that when contact tracing is fully utilized, one can mitigate the eruption of the pandemic by breaking the current chain of proliferation of the corona virus, and thus helping to reduce the rate of recent epidemics [16] . Topological indices of a chemical compound are molecular descriptors. Several topological indices have been defined and utilised in QSPR/QSAR studies to understand the relationship between molecular structure and potential physicochemical properties [16] [17] [18] [19] . Stimulated by such varied applications of biological networks, we have in this paper obtained a number of topological indices of these networks and we have also verified the expressions using computer codes. In this process we have also identified several logical flaws found in Gao et al. [20] , in the computation of certain topological indices for some of the trees and we have provided corrected expressions. The distance d (u,v) between two vertices u and v in a connected graph G is the length of the shortest path between them. If l ≥ 1, then the set {1, … , l} will be denoted by [l] . For any vertex v ∈ V(G) , the eccentricity of v is defined as Ghorbani and Khaki [18] introduced the eccentric version of geometric-arithmetic index as fourth geometric-arithmetic eccentricity index which is stated as Further, the fourth Zagreb index, the fourth multiplicative Zagreb index, the sixth Zagreb index, the sixth multiplicative Zagreb index, and the fourth and sixth Zagreb polynomial index [19, 20] are defined as follows: In addition, the fifth multiplicative atom bond connectivity index [20] is defined by . x (s)+ (t) ; . A hypertree HT(l) [21] is a complete binary tree T l (a binary tree with l levels where level a, 0 ≤ a ≤ l contains 2 a vertices) and the vertices of T l are labeled as follows: The root node has label 1 and is said to be at level 0. The children of the vertex x are labeled with 2x and 2x + 1. Additional edges in a hypertree are horizontal, where two vertices in the same level a, 1 ≤ a ≤ l , are joined by an edge if their label difference is 2 a−1 , see Fig. 1 . Gao et al. [20] have obtained various topological indices defined above in the form of Theorem 1 for the hypertree, HT(l) with l-levels. As their results are erroneous as shown here, we provide the corrected results for various topological indices reported by Gao et al. [20] . Consider the hypertree HT (3) in Fig. 1 as an example to derive the correct expressions for HT. By manual calculation of the eccentricities for the various vertices in The expression obtained manually in Eq. (1) is in disagreement with Eq. (3) obtained by the application of the expression derived for Zg 4 (HT(l)) by Gao et al. [20] . Similarly, Eqs. (2) and (4) imply that the expression obtained in Ref [20] for Π * 4 (HT(l)) is also not correct. More comparisons are given in Table 1 for all of the indices for the hypertrees. We have also used TopoChemie-2020 [22] , a suite of Fortran'95 codes to compute all of the topological indices considered here to validate the results. The results obtained from TopoChemie-2020 are shown in Table 1 for comparison with the results obtained from present work and those from Ref [20] . (4) Π * 4 (HT(3)) = 6(3) x(4(4) + 2)(2(4) + 2)] = 23, 224, 320 Proceeding along the same lines, we prove the remaining equations. □ In this section, we compute certain distance and degree based topological indices of hypertrees that have not been obtained before. We begin with the following definitions. where d i = d G v i is denoted by the degree of vertex v i for i =1,2,…,n such that d 1 ≥ d 2 ≥ ⋯ ≥ d n . The second Zagreb index M 2 (G) [23] of graph G is defined as ) be a graph with n = |V(G)| vertices and m = |E(G)| edges. The edge connecting the vertices i and j is denoted by ij. Then Atom-bond Connectivity Index (ABC) [24] is defined as Proof First, we partition the edge set of HT(l) as follows: Proof First, we partition the vertex set of HT(l) as follows: By the definition, we have Fig. 2 . Note that the number of edges between level i and i + 1, For example, the hypertree HT(4 ) with n(v) = 14 is shown in Fig. 2 . Note that the number of edges between level i and i + 1, It is easy to verify that n(u) = 1 and n(v) = 2 l − 1 , when (uv) ∈ B e , u is in level 0 and v is in level 1. = 2(1)(2 l − 1) + l−1 ∑ i=1 (n(u)n(v))2 i+1 + (2 l − 1)(n(u)n(v)) = 2(1)(2 l − 1) + l−1 ∑ i=1 2 i+1 (2 l+1 − 2 l−i+1 + 1) 2 l−i+1 − 2 + 2 l − 1 (2 l − 1) 2 = 2 3l − 27 × 2 2l + 12l × 2 l + 21 × 2 l + 8l × 2 2l + 5. Proof Since HT(l) is bi-regular with degree 2 and degree 4 vertices, we partition the vertices (ordered pairs) of HT(l) as follows: = 2(1 + (2 l − 1)) + l−1 ∑ i=1 (n(u) + n(v))(2 i+1 ) + (2 l − 1)(n(u) + n(v)) = 2 l+1 + l−1 ∑ i=1 (2 l+1 )[(2 l+1 − 2 l−i+1 + 1) + 2 l−i+1 − 2 ] + (2 l − 1)((2 l − 1) + (2 l − 1)) = 6 × (2 l − 1) 2 . ◻ S(HT(l)) = 38l × 2 l − 70 × 2 2l + 50 × 2 l + 24l × 2 2l + 20 S(HT(l)) = ∑ (u,v)∈V 22 (d(u) + d(v))d(u, v) + ∑ (u,v)∈V 24 (d(u) + d(v))d(u, v) + ∑ (u,v)∈V 44 (d(u) + d(v))d(u, v) = l × 2 l+2 + 2 l+1 l−1 ∑ i=1 2 i−1 (4i + 1) + 1 + (2 + 4) i−1 ∑ i=2 i−1 ∑ j=1 (l − i + 2j)2 l+j−1 + l−1 ∑ i=1 (l − i)2 l + l−1 ∑ i=2 i−1 ∑ j=1 (l − i + 2j + 1)2 l+j−1 + l−1 ∑ i=1 (l − i + 1)2 l + l−1 ∑ i=1 i × 2 i + (4 + 4) l−2 ∑ k=1 l−1−k ∑ i=2 i−1 ∑ j=1 (l − i + 2j − k)2 l+j−k−1 + l−2 ∑ k=1 l−1−k ∑ i=1 (l − i − k)2 l−k + l−2 ∑ k=1 l−1−k ∑ i=2 i−1 ∑ j=1 (l − i + 2j − k + 1)2 l+j−k−1 + l−2 ∑ k=1 l−1−k ∑ i=1 (l − i − k + 1)2 l−k + l−1 ∑ i=1 i−1 ∑ j=1 2 j−1 × 2j + i−1 ∑ j=1 2 j−1 × (2j + 1) + 1 2 i−1 = 38l × 2 l − 70 × 2 2l + 50 × 2 l + 24l × 2 2l + 20. ◻ Proof Since HT(l) is bi-regular with degrees 2 and 4, we partition the vertices (ordered pairs) of HT(l) as follows: The results obtained from TopoChemie-2020 are shown in Table 2 for comparison with the results obtained from Theorems 4.1.1-4.1.7. In this section, we consider the topological indices of hypertrees generated through corona products that have not been considered before. Definition 5.1 [29] The corona product G 1 ⊙ G 2 of two graphs G 1 with n 1 vertices and m 1 edges and G 2 with n 2 vertices and m 2 edges is defined as the graph obtained Gut(HT(l)) = 60l × 2 l − 111 × 2 2l + 80 × 2 l + 36l × 2 2l + 32. by taking one copy of G 1 and n 1 copies of G 2 , and then joining the ith vertex of G 1 with an edge to every vertex in the ith copy of G 2 . It follows from the definition of the corona that G 1 ⊙ G 2 has n 1 + n 1 n 2 vertices and m 1 + n 1 m 2 + n 1 n 2 edges. It is easy to see that G 1 ⊙ G 2 is not in general isomorphic to G 2 ⊙ G 1 . For illustration, the corona product of hypertree HT(3) and path P 3 is given in Fig. 3 . Let HT(l) ⊙ P r , l, r ≥ 2 be the corona product of hypertree and a path network on r vertices. Then we have the following: Zg 4 HT(l) ⊙ P r = 6(l + 1) + (r + 2) 2 l (4l − 3) − 4l + 2 + 2 l (r)(4l + 1) + r(2l + 3) Zg 6 HT(l) ⊙ P r = 3(l + 1) 2 + (r + 2) 2 l + 2.2 l 2l 2 − 3l + 2 − 2l 2 − 4 + 2 l r 4l 2 + 2l + r(l + 1)(l + 2) + (r − 1)(4 × 2 l − 2l 2 + 8 × 2 l l 2 − 4) (l + a) 2 + l + a 2 a (2+r) × 4l 2 + 2l 2 a x 2(l+a+1) + (r − 1)x 2(l+2) ; From the definitions of eccentricity-based topological indices, we get 2 a (l + a + 1) 2 + (r − 1)(l + 2) 2 = 3(l + 1) 2 + (r + 2) 2l + 2 × 2 l 2l 2 − 3l + 2 − 2l 2 − 4 + 2 l r 4l 2 + 2l + r(l + 1)(l + 2) + (r − 1) 4 × 2 l − 2l 2 + 8 × 2 l l 2 − 4 (l + a + 1) 2 a+1 × (l + 2) 2(r−1) ; Theorem 5.3 Let HT(k) ⊙ P n , k, n ≥ 2 be the corona product of hypertree and a path network on n vertices. Then the atom bond connectivity index Proof First, we partition the edge set of HT(k) ⊙ P n as follows: 2 a x 2(l+a+1) + (r − 1)x 2(l+2) ; Remark 5. 6 The results obtained from TopoChemie-2020 are shown in Table 3 for comparison with the results obtained from Theorems 5.2-5.5. The recursive hypertree networks considered in this study could have several applications to chemical problems such as recursive molecular networks that occur in chemical applications, for example, dendrimers [30] . The QSAR studies of dendrimers could then be benefited by the various topological indices derived in this work for hypertrees. Furthermore, recent extension of topological indices to relativistic topological indices considered in [31] could have significant ramifications in the prediction of relativistic chemical properties of complex network of molecular and material systems when heavier atoms such as Pt, Pd and other heavy elements are present [32] [33] [34] [35] . For such systems relativistic effects are quite important [32] [33] [34] [35] . It is anticipated that when the expressions derived here for the topological indices of hypertrees are generalized to encompass such relativistic corrections [32] [33] [34] [35] then the generalized relativistic expressions could have important applications in the prediction of topological properties of dendrimeric metal-organic networks containing very heavy atoms. Moreover, graph products such as the corona products and other products have been recently considered for the computation of Mostar indices of nanomaterials [36] , and thus we expect more research along these lines employing graph products in the future. Biological processes are well interpreted by reliable models for interaction networks. The structure of social networks has to be studied along with the spread of infectious diseases like corona virus [8] . In this paper we have considered certain biochemical networks such as k-level hypertrees and corona product of hypertrees and several topological indices have been obtained for the same. Table 3 Results obtained for the corona product of hypertree and a path network with n vertices HT(k) ⊙ P n with the computer code compared with the results from the expressions in Theorems 5. Be-CoDiS: a mathematical model to predict the risk of human diseases spread between countries. Validation and application to the 2014 ebola virus disease epidemic Understanding Infectious Disease Dynamics. Princeton Series in Theoretical and Computational Biology A mathematical model for assessing control strategies against West Nile virus Early dynamics of transmission and control of COVID-19: a mathematical modelling study Real-time forecasts of the COVID-19 epidemic in china from February 5th to February 24th Artificial intelligence for infectious disease big data analytics Risk-Aware Identification of Highly Suspected COVID-19 Cases in Social IoT: A Joint Graph Theory and Reinforcement Learning Approach Graph Theory and Networks in Biology A guide to conquer the biological network era using graph theory Networks and the epidemiology of infectious disease Gene expression networks Bipartite graphs in systems biology and medicine: a survey of methods and applications Predicting protein functions from redundancies in large-scale protein interaction networks Modelling disease outbreaks in realistic urban social networks Normal and pathological oscillatory communication in the brain Applications of machine learning and artificial intelligence for Covid-19 (SARS-CoV-2) pandemic: a review Some distance-based topological indices of a non-commuting graph A note on the fourth version of geometric-arithmetic index Fourth zagreb index of circumcoronene series of benzenoid. Leonardo El Study of biological networks using graph theory A multiprocessor interconnection topology TopoChemie-2020, Fortran'95 codes for the computation of topological indices Graph theory and molecular orbitals: total π-electron energy of alternant hydrocarbons On atom-bond connectivity index On a novel structural descriptor PI A formula for the Wiener number of trees and its extension to graphs containing cycles Topological organic chemistry: graph theory and topological indices of alkanes Selected properties of the Schultz molecular topological index The corona between cycles and paths Nested wreath groups and their applications to phylogeny in biology and Cayley trees in chemistry and physics Relativistic topological molecular descriptors of metal trihalides Relativistic Effects in Chemistry: Part A Theory & Techniques Potential-energy surfaces for Pt 2 + H and Pt + H interactions Ten Low-lying electronic states of Pd 3 A comparative theoretical study of bonding in UO ++ 2 , UO + 2 , UO 2 , UO − 2 , OUCO, O 2 U(CO) 2 and UO 2 CO 3 Edge Mostar index of chemical structures and nanostructures using graph operations Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations The work of R. Sundara Rajan is supported by Project No. ECR/2016/1993, Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India. Further, we thank G. Kirithiga Nandini, Department of Computer Science and Engineering, Hindustan Institute of Technology and Science, India, for her fruitful suggestions. The authors sincerely thank the reviewers for their comments and suggestions which significantly improved the quality of this paper. Authors wish to thank Prof M. Arockiaraj for his help in checking the expressions. By the definition, we have Theorem 5. 4 Let HT(k) ⊙ P n , k, n ≥ 2 be the corona product of hypertree and a path network on n vertices. Then the first Zagreb indexProof First, we partition the vertex set of HT(k) ⊙ P n as follows:By the definition, we have Proof First, we partition the edge set of HT(k) ⊙ P n as follows:HT(k) ⊙ P n � �d(u) = n + 4 and d(v) = 3 � Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper.