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For a given graph G, the second Zagreb eccentricity index ξ 2 (G) is defined as the product of the eccentricities of two adjacent vertex pairs in G. This paper mainly studies the problem of determining the graphs that minimize the second Zagreb eccentricity index among n-vertex bipartite graphs with a fixed number of edges and diameter. To be specific, we determine the sharp lower bound on the second Zagreb eccentricity index over the bipartite graphs of order n in terms of fixed edges and diameter. The extremal graphs attaining these lower bounds are fully characterized. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]

Identifying certain conditions that ensure the Hamiltonicity of graphs is highly important and valuable due to the fact that determining whether a graph is Hamiltonian is an NP-complete problem.For a graph G with vertex set V(G) and edge set E(G), the first Zagreb index ( M 1 ) and second Zagreb index ( M 2 ) are defined as M 1 (G) = ∑ v i v j ∈ E (G) (d G (v i) + d G (v j)) and M 2 (G) = ∑ v i v j ∈ E (G) d G (v i) d G (v j) , where d G (v i) denotes the degree of vertex v i ∈ V (G) . The difference of Zagreb indices ( Δ M ) of G is defined as Δ M (G) = M 2 (G) - M 1 (G) .In this paper, we try to look for the relationship between structural graph theory and chemical graph theory. We obtain some sufficient conditions, with regards to Δ M (G) , for graphs to be k-hamiltonian, traceable, k-edge-hamiltonian, k-connected, Hamilton-connected or k-path-coverable. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. The Sombor index of a graph G is defined as S O (G) = ∑ u v ∈ E (G) d G 2 (u) + d G 2 (v) , where d G (u) denotes the degree of u in G. Let B n d be the set of all bipartite graphs of diameter d with n vertices. In this paper, we determine the sharp upper bound on the Sombor index of G ∈ B n d . In addition, we propose an algorithm for searching the largest Sombor index among B n d . Furthermore, the relationship between the maximal Sombor index of G ∈ B n d and the diameter d is established. Finally, we obtain the largest, the second-largest, the third-largest and the smallest Sombor indices of bipartite graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. Denote by d i the degree of a vertex v i in G. Let f (x , y) > 0 be a real symmetric function. Consider an edge-weighted graph in such a way that for each edge v i v j of G, the weight of v i v j is equal to the value f (d i , d j) . Therefore, we have a degree-based weighted adjacency matrix A f (G) of G, in which the (i, j)-entry is equal to f (d i , d j) if v i v j is an edge of G and is equal to zero otherwise. Let x be a positive eigenvector corresponding to the largest eigenvalue λ 1 (A f (G)) of the weighted adjacency matrix A f (G) . In this paper, we first consider the unimodality of the eigenvector x on an induced path of G. Second, if f(x, y) is increasing in the variable x, then we investigate how the largest weighted adjacency eigenvalue λ 1 (A f (G)) changes when G is perturbed by vertex contraction or edge subdivision. The aim of this paper is to unify the study of spectral properties for the degree-based weighted adjacency matrices of graphs. [ABSTRACT FROM AUTHOR]

The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then its relationship with other graph invariants has been studied by several authors. In this paper we obtain new inequalities involving the inverse degree index, and we characterize graphs which are extremal with respect to them. In particular, we obtain several inequalities relating the inverse degree index with the first and second Zagreb indices, the general first and second Zagreb indices, the Forgotten index, the general sum-connectivity index, the Sombor index and the misbalance indeg index, and several parameters of the molecular graph as the number of vertices, the number of edges, the minimum degree and the maximum degree. Also, we compute the inverse degree index for some classes of chemical graphs. Furthermore, some applications are given to the study of the physicochemical properties of three classes of compounds: polyaromatic hydrocarbons, polychlorobiphenyls, and octane isomers. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a new graph-theoretic invariant called the bi-Wiener index. The bi-Wiener index W b (G) of a bipartite graph G is defined as the sum of all (shortest-path) distances between two vertices from different parts of the bipartition of the vertex set of G. We start with providing a motivation connected with the potential uses of the new invariant in the QSAR/QSPR studies. Then we study its behavior for trees. We prove that, among all trees of order n ≥ 4 , the minimum value of W b is attained for the star S n , and the maximum W b is attained at path P n for even n, or at path P n and B n (2) for odd n where B n (2) is a broom with maximum degree 3. We also determine the extremal values of the ratio W b (T n) / W (T n) over all trees of order n. At the end, we indicate some open problems and discuss some possible directions of further research. [ABSTRACT FROM AUTHOR]

One of the most important information related to molecular graphs is given by the determination (when possible) of upper and lower bounds for their corresponding topological indices. Such bounds allow to establish the approximate range of the topological indices in terms of molecular structural parameters. The purpose of this paper is to provide new inequalities relating several classes of variable topological indices including the first and second general Zagreb indices, the general sum-connectivity index, and the variable inverse sum deg index. Also, upper and lower bounds on the inverse degree in terms of the first general Zagreb are found. Moreover, the characterization of extremal graphs with respect to many of these inequalities is obtained. Finally, some applications are given. [ABSTRACT FROM AUTHOR]

A nonnegative integer p is realizable by a graph-theoretical invariant I if there exists a graph G such that I (G) = p . The inverse problem for I consists of finding all nonnegative integers p realizable by I. In this paper, we consider and solve the inverse problem for the Mostar index, a recently introduced graph-theoretical invariant which attracted a lot of attention in recent years in both the mathematical and the chemical community. We show that a nonnegative integer is realizable by the Mostar index if and only if it is not equal to one. Besides presenting the complete solution to the problem, we also present some empirical observations and outline several open problems and possible directions for further research. [ABSTRACT FROM AUTHOR]

In this paper, we examine the Gutman–Milovanović index and establish new upper and lower bounds for it. These bounds include terms related to the general sum connectivity index, the general second Zagreb index, and the hyperbolicity constant of the underlying graph. Also, we model physicochemical properties of polyaromatic hydrocarbons using the Gutman–Milovanović index. [ABSTRACT FROM AUTHOR]

The global pandemic caused by the novel virus SARS-CoV-2 (Severe Acute Respiratory Syndrome CoronaVirus 2), also known as COVID-19, is now a serious public health concern that has affected people worldwide. The condition has become worse due to a lack of adequate treatment. To combat the pandemic, several drugs are being investigated. A topological index (or molecular descriptor) is a numerical parameter that correlates the molecular structure of a chemical compound to its various physico-chemical properties and plays a significant role in the development of QSPR/QSAR (quantitative structure–property relationship/quantitative structure-activity relationship) models. In this study, we evaluate the degree-based topological indices (namely, the Nirmala index, first and second inverse Nirmala indices, geometric-quadratic and quadratic-geometric indices) of nine antiviral drugs (namely, Molnupiravir, Remdesivir, Chloroquine, Ritonavir, Theaflavin, Arbidol, Hydroxychloroquine, Thalidomide and Lopinavir) used in the remedy of COVID-19 patients, with the help of their respective M-polynomials. Also, we calculate the neighborhood degree sum-based indices of Molnupiravir by using its neighborhood M-polynomial (that is, NM-polynomial). In addition, we execute the correlation analysis among the topological indices and physico-chemical properties of these antiviral drugs. Furthermore, we demonstrate the QSPR models for strong correlation through the linear, quadratic and cubic regression analysis to appraise the effectiveness of the topological indices. And, the squared correlation coefficients obtained from the performed curvilinear regression models are compared with those acquired in the previous studies. The obtained topological indices and established QSPR models which may be helpful to predict the pharmacokinetic properties of these antiviral drugs and in the discovery of new drugs related to the medication for the COVID-19 pandemic. [ABSTRACT FROM AUTHOR]

I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by S O 1 , S O 2 , ... , S O 6 . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal values of the graph invariants S O 5 and S O 6 among all trees and molecular trees of order n, and characterize the trees and molecular trees that achieve the extremal values, respectively. Thus, the problem is completely solved. [ABSTRACT FROM AUTHOR]

The first inverse Nirmala index is a novel degree-based topological descriptor that was introduced in 2021. Preliminary QSPR investigations suggest that this index deserves further consideration because of its unusually good predictive potential. This paper investigates the relations between this index with some elementary graph quantities and some related degree-based topological index. Further, the computational analysis will reveal extremal graphs among trees, molecular trees, all connected graphs, and their molecular counterparts. [ABSTRACT FROM AUTHOR]

Sierpiński graphs are frequently related to fractals, and fractals apply in several fields of science, i.e., in chemical graph theory, computer networking, biology, and physical sciences. Functions and polynomials are powerful tools in computer mathematics for predicting the features of networks. Topological descriptors, frequently graph constraints, are absolute values that characterize the topology of a computer network. In this essay, Firstly, we compute the M-polynomials for Sierpiński-type fractals. We derive some degree-dependent topological invariants after applying algebraic operations on these M-polynomials. [ABSTRACT FROM AUTHOR]

Topological indices are obtained from molecular graphs and are real numbers that can forecast the biological and physicochemical properties of several anti-cancer treatments, including skin cancer, breast cancer, and blood cancer. This article focuses on the application of topological indices in predicting the effectiveness of several drugs used to cure blood cancer, such as Pamidronic acid, Alpelisib, Prednisone, Olaparib, Ribociclib, Tucatinib, dexamethasone, docetaxel, Midostaurin, paclitaxel, toremifene, and venetoclax. The article investigates the mathematical relationships between physical and chemical qualities and data encoded in chemical structures under characteristics such as molecular weight, molar volume, and complexity. Several topological indices are used in this context to forecast the physicochemical characteristics of the drugs. [ABSTRACT FROM AUTHOR]