Hosoya Index Research Articles

Enumeration of the Hosoya index of pericondensed benzenoid system

In chemical graph theory, topological indices play an important role and have various uses in quantitative structure-property relationships (QSPR) as well as quantitative structure-activity relationships (QSARs). The Hosoya index is one of them, performing a molecular descriptor in mathematical chemistry. Therefore, progressing with investigations requires computing the Hosoya index of distinct molecular graphs. This research presents a computational method for determining the Hosoya index of pericondensed benzenoid systems using the transfer matrix approach and the Hosoya vector.

Matchings in bipartite graphs with a given number of cuts

A matching in a graph is a set of pairwise nonadjacent edges. Denote by m(G,k) the number of matchings of cardinality k in a graph G. A quasi-order ⪯ is defined by G⪯H whenever m(G,k)≤m(H,k) holds for all k. Let BG1(n,γ) be the set of connected bipartite graphs with n vertices and γ cut vertices, and BG2(n,γ) be the set of connected bipartite graphs with n vertices and γ cut edges. We determine the greatest and least elements with respect to this quasi-order in BG1(n,γ) and the greatest element in BG2(n,γ) for all values of n and γ. As corollaries, we find that these graphs maximize (resp. minimize) the Hosoya index and the matching energy within the respective sets.

The matching polynomial of the path-tree of a complete graph

Let Tn denote the path-tree of the complete graph Kn, we show that the matching polynomial μ(Tn,x) of Tn is equal toμ(Kn,x)∏i=2n−1μ(Ki,x)∏j=i−1,j≠in−1j.This result has a number of consequences. It follows that the roots of μ(Tn,x) (or the eigenvalues of Tn), the (matching) energy of Tn, and the Hosoya index of Tn can all be obtained directly from that of complete graphs Ki(1≤i≤n).

The role of physicochemical and topological parameters in drug design

Quantitative structure activity relationship (QSAR) is a widely used tool in rational drug design that establishes relationships between the physicochemical and topological descriptors of ligands and their biological activities. Obtained QSAR models help identify descriptors that play pivotal roles in the biological activity of ligands. This not only helps the prediction of new compounds with desirable biological activities but also helps with the design of new compounds with better activities and low toxicities. QSAR commonly uses lipophilicity (logP), hydrophobicity (logD), water solubility (logS), the acid–base dissociation constant (pKa), the dipole moment, the highest occupied molecular orbital (HOMO), the lowest unoccupied molecular orbital (LUMO), molecular weight (MW), molar volume (MV), molar refractivity (MR), and the kappa index as physicochemical parameters. Some commonly used topological indices in QSAR are the Wiener index, Platt index, Hosoya index, Zagreb indices, Balaban index, and E-state index. This review presents a brief description of the significance of the most extensively used physicochemical and topological parameters in drug design.

Computing the Hosoya Index of Some Nanostar Dendrimers

Dendrimers are highly branched macromolecules built up from a monomer, with new branches added in steps until a tree structure is created. The various biological characteristics of dendrimers are a good choice in chemistry, biology, the medical field, and nano-science. A topological index is a type of molecular descriptor that is calculated based on the molecular graph of a chemical structure. The Hosoya index is a well-known topological index that is used to predict some of their physico-chemical properties from the structure of molecules. The Hosoya index of a graph is defined as the total number of its independent edge sets. In this paper, we study the Hosoya index of some families of nanostar dendrimers. We obtain the formulas of the Hosoya index for two infinite classes of dendrimers, namely, nanostar dendrimer and tetrathiafulvalene dendrimer. We use the Mathematica program to evaluate the results and accuracy of calculations. Our results can be used in analyzing the molecular topology of these families of nanostar dendrimers.

Extremal Bicyclic Graphs with Respect to Permanental Sums and Hosoya Indices

Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the absolute value of all coefficients for the matching polynomial. And the permanental sum of a graph is the sum of the absolute value of all coefficients of the permanental polynomial. In this paper, we characterize the second to sixth minimal Hosoya indices of all bicyclic graphs. Furthermore, using the results, the second to sixth minimal permanental sums of all bicyclic graphs are also characterized.

Solution to a Conjecture on the Permanental Sum

Let G be a graph with n vertices and m edges. A(G) and I denote, respectively, the adjacency matrix of G and an n by n identity matrix. For a graph G, the permanent of matrix (I+A(G)) is called the permanental sum of G. In this paper, we give a relation between the Hosoya index and the permanental sum of G. This implies that the computational complexity of the permanental sum is NP-complete. Furthermore, we characterize the graphs with the minimum permanental sum among all graphs of n vertices and m edges, where n+3≤m≤2n−3.

Matching Polynomials and Independence Polynomials of Benzenoid Chains

Benzenoid chains, also known as hexagonal chains, are a class of organic compounds that consist of an arrangement of hexagonal rings fused together. In this article, we first present reduction formulas to compute the matching polynomial and independence polynomial of any benzenoid chain by utilizing the transfer matrix technique. Subsequently, computational formulas for the Hosoya index and Merrifield-Simmons index of benzenoid chains are derived. Furthermore, the expected values of the Hosoya index and Merrifield-Simmons index for random benzenoid chains are also obtained.

On the Hosoya Index and Permanental Sum of Rose Graphs

On the Hosoya Index and Permanental Sum of Rose Graphs

The Expected Values of Hosoya Index and Merrifield–Simmons Index of Random Hexagonal Cactus Chains

Hosoya index and Merrifield-Simmons index are two well-known topological descriptors that reflex some physical properties, such as boiling points and heat of formation, of benzenoid hydrocarbon compounds. In this paper, we establish the generating functions of the expected values of these two indices of random hexagonal cacti. This generalizes the results of Doslic and Maloy, published in Discrete Mathematics in 2010. By applying the ideas on meromorphic functions and the growth of power series coefficients, the asymptotic behaviors of these indices on the random cacti have been established.

Hosoya index of thorny polymers

A matching in a graph G is a collection of edges of G such that no two of them share a vertex. The number of all matchings in G is called its Hosoya index. In this paper, we compute Hosoya indices of several classes of unbranched polymers made of cycles of the same lengths arranged around a middle path and decorated by attaching to each vertex, a given number of pendent vertices or thorns. We establish linear recurrences satisfied by those numbers and obtain explicit formulas in terms of Fibonacci polynomials and their generalizations. Some possible directions of future research are also indicated.

Counting the maximal and perfect matchings in benzenoid chains

Došlić et al. obtained general generating functions for the numbers of maximal matchings in three classes of benzenoid chains. By using the Hosoya vector and k-matching vector, Cruz et al. and Oz et al. researched the Hosoya index and the k-matching number of benzenoids, respectively. Inspired by these results, by using the maximal matching vector, we show that the number of maximal matchings of a benzenoid chain with n hexagons equals to the product of n certain matrices. In addition, we obtain the numbers of perfect matchings of all benzenoid chains.

The expected values of the total numbers of independent edge sets and independent sets in random alpha-type pentagonal chains

A independent edge set of G containing mutually independent edges is also called a matching of G. The total numbers of matchings and independent sets of a graph G, namely, the Hosoya index and the Merrifield-Simmons index, respectively, are two important topological indices. We compute the average total numbers of independent edge sets and independent sets in random alpha-type pentagonal chains.

On diameter $5$ trees with the maximum number of matchings

A matching in a graph is any set of edges of this graph without common vertices. The number of matchings, also known as the Hosoya index of the graph, is an important parameter, which finds wide applications in mathematical chemistry. Previously, the problem of maximizing the Hosoya index in trees of radius $2$ (that is, diameter $4$) of fixed size was completely solved. This work considers the problem of maximizing the Hosoya index in trees of diameter $5$ on a fixed number $n$ of vertices and solves it completely. It turns out that for any $n$ the extremal tree is unique. Bibliography: 6 titles.

The Minimum Hosoya Index of a Kind of Tetracyclic Graph

The Minimum Hosoya Index of a Kind of Tetracyclic Graph

Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups.

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph of a finite group is a graph where non-central elements of are its vertex set, while two different elements are edge connected when they do not commute in . In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of . We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

Computing the Number of Matchings in Catacondensed Benzenoid Systems

The Hosoya index of is defined as the total number of independent edge sets (number of -matchings ) in . The Hosoya index is one of the most important topological indices in the field of mathematical chemistry because of its relationship with several thermodynamic properties. Therefore, computation of the number of -matchings of various molecular structures has importance. Two methods, one for computing the number of the Hosoya index of catacondensed benzenoid systems and the other for the number of -matchings in benzenoid chains (unbranched catacondensed benzenoid systems), have been presented so far. In this paper, a method based on some transfer matrices to compute the number of -matchings of arbitrary (both unbranched and branched) catacondensed benzenoid systems is presented. Moreover, some algorithms are designed to keep the applicability of the method the same as increases.

Hosoya index of VDB-weighted graphs

Given a graph G = ( V , E ) with vertex set V and edge set E , we extend the concept of k -matching number and Hosoya index to a weighted graph ( G ; ω ) , where ω is a weight function defined over E . In particular, if φ is a vertex-degree-based (VDB) topological index defined via φ = φ ( G ) = ∑ u v ∈ E φ d G ( u ) , d G ( v ) , where d G ( u ) is the degree of the vertex u and φ i , j is an appropriate function with the property φ i , j = φ j , i , then we consider the weighted graph ( G ; φ ) with weight function φ : E → R defined as φ ( u v ) = φ d G ( u ) , d G ( v ) , for all u v ∈ E . It turns out that m ( ( G ; φ ) , 1 ) , the number of weighted 1-matchings in ( G ; φ ) , is precisely φ ( G ) , and for k ≥ 2 , the k -matching numbers m ( ( G ; φ ) , k ) can be viewed as new k th order VDB-Hosoya indices. Later, we consider the extremal value problem of the Hosoya index over the set T n ; φ = ( T ; φ ) : T ∈ T n , where T n is the set of trees with n vertices.

The Expected Value of Hosoya Index and Merrifield–Simmons Index in a Random Cyclooctylene Chain

The Hosoya index m(G) and the Merrifield–Simmons index i(G) of a graph G are the number of matchings and the number of independent sets in G. In this paper, we establish exact formulas for the expected value of the Hosoya index and Merrifield–Simmons index of the random cyclooctylene chains, which are graphs of a chemical chain consisting of n octagons, each of which is connected to the end of the previous octagon by an edge. In addition, we obtain the expected values and the average values of the two indexes through the relevant chemical diagrams and a series of accurate formulas with respect to the set of all cyclooctylene chains with n octagons.

A New Proof of a Result Concerning a Complete Description of $$ (n, n + 2) $$-Graphs with Maximum Value of the Hosoya Index

A New Proof of a Result Concerning a Complete Description of $$ (n, n + 2) $$-Graphs with Maximum Value of the Hosoya Index