Benzenoid Chains Research Articles

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.

The Multivariable Zhang–Zhang Polynomial of Phenylenes

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons.

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.

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.

On Bond Incident Connection Indices of Polyomino and Benzenoid Chains

The problem of estimating physicochemical properties of chemical structures is a well-investigated problem in chemistry. A number of approaches have been proposed to solve this certain problem. One of these approaches is linked with molecular descriptors. The molecular descriptors that are defined in terms of connection numbers of adjacent vertices of the chemical graph of the considered chemical structure are known as bond incident connection (BIC) indices. The primary aim of this article is to study BIC indices of polyomino and benzenoid chains.

Computing the Number of k-Matchings in Benzenoid Chains

Computing the Number of k-Matchings in Benzenoid Chains

Enumeration of Independent Sets in Benzenoid Chains

Enumeration of Independent Sets in Benzenoid Chains

Omega, Sadhana, Theta, and PI Polynomials of Double Benzonoid Chain

Counting polynomials are closely related to certain features of chemical graphs and provide an elegant means of expressing graph topological invariants. The current paper aims to calculate four polynomials for double benzenoid chains, Sadhana, omega, theta, and Padmakar–Ivan (PI). The edge‐cut method is used to derive analytical closed expressions for these polynomials.

The generalized Zhang–Zhang polynomial of benzenoid systems – theory and applications

The generalized Zhang–Zhang (GZZ) polynomial was introduced recently aiming to increase the sensitivity of the well-known Zhang–Zhang polynomial onto π-electron cyclic conjugation of 10-membered rings. Here, the recursive formulas for the calculation of the GZZ of the benzenoid systems are derived. Then, an algorithm for calculating the GZZ of benzenoid chains is given. Lastly, testing the chemical applicability of GZZ is performed.

Computing the Merrifield-Simmons indices of benzenoid chains and double benzenoid chains

Computing the Merrifield-Simmons indices of benzenoid chains and double benzenoid chains

The Tutte polynomials of catacondensed benzenoid systems

A catacondensed benzenoid system (resp. benzenoid chain) is a benzenoid system whose inner dual graph is a tree (resp. a path). The Tutte polynomial of a graph is a two-variable polynomial whose evaluations at various points are equivalent to the exact solutions of many counting problems. In this paper, we introduce a graph vector at a given edge which related to the Tutte polynomial. Based on this concept and by three classes transfer matrices, we get the reduction formula for Tutte polynomial of any catacondensed benzenoid system. Moreover, the number of spanning trees for any catacondensed benzenoid system is also determined via a product of $$(2\times 2)$$ matrices with entries in N. As a by-product, we study the extremum problem of the number of spanning trees over the set of cataconsed hexagonal systems with one branched hexagon.

Maximal matchings in polyspiro and benzenoid chains

A matching M of a graph G is maximal if it is not a proper subset of any other matching in G. Maximal matchings are much less known and researched than their maximum and perfect counterparts. In this paper we present the recurrences and generating functions for the sequences enumerating maximal matchings in two classes of chemically interesting linear polymers: polyspiro chains and benzenoid chains. We also analyze the asymptotic behavior of those sequences and determine the extremal cases.

Omega, Sadhana, and PI Polynomials of Quasi-Hexagonal Benzenoid Chain.

Counting polynomials are important graph invariants whose coefficients and exponents are related to different properties of chemical graphs. Three closely related polynomials, i.e., Omega, Sadhana, and PI polynomials, dependent upon the equidistant edges and nonequidistant edges of graphs, are studied for quasi-hexagonal benzenoid chains. Analytical closed expressions for these polynomials are derived. Moreover, relation between Padmakar–Ivan (PI) index of quasi-hexagonal chain and that of corresponding linear chain is also established.

On the edge eccentric and modified edge eccentric connectivity indices of linear benzenoid chains and double hexagonal chains

Let G be a molecular graph. The edge eccentric connectivity index of G, ξec(G), is defined as ξec(G)=∑f∈E(G)deg(f).ecc(f), where deg(f) denotes the degree of edge f and ecc(f) is the largest distance between f and any other edge g of G. The modified edge eccentric connectivity index of G, Λec(G), is defined as Λec(G)=∑f∈E(G)S(f).ecc(f), where S(f) denotes the sum of the degrees of edges neighborhoods of edge f. In this paper, exact formulas for the edge eccentric connectivity index and modified edge eccentric connectivity index of linear benzenoid chains and double hexagonal chains were computed.

EMD of Linear Benzenoid Chain Using Chemical Graph

Let Gm be a undirected graph. Having a Molecule, atoms indicate by vertices and bonds indicate by edges. The main aim of this paper explains Extended Medium Domination on Linear Benzenoid Chain like Benzene, Naphthalene, Anthracene and their generalisation will be proved.

Two physicochemical properties of benzenoid chains: solvent accessible molecular volume and molar refraction

Predicting physicochemical properties of molecules is one of the fundamental tasks in chemical physics. Many predictive methods have been developed for correlating the molecular structures with their physicochemical properties. One of the simplest such methods involves topological indices. Edge connectivity index (or equivalently, reformulated Randić index), which is denoted as ε, seems to be a good topological index for predicting the solvent accessible molecular volume and molar refraction of polycyclic aromatic hydrocarbons. In this paper, a closed-form formula for calculating the reformulated Randić index ε of benzenoid hydrocarbon chains (or simply, benzenoid chains, which represent a type of polycyclic aromatic hydrocarbons) is derived. Benzenoid chains with maximum (and minimum) ε value are also determined from the collection of all benzenoid chains having fixed number of hexagonal rings. Moreover, an attempt is made to generalize the obtained results for reformulated bond incident degree indices.

Extremal Even Polygonal Chains on Wiener Numbers

Denote by the set of h-polygonal chains (where h is even) with n congruent regular h-polygons (). For any let be the Wiener number of In this paper, we show that with the equalities on the left holding only if and the equalities on the right holding only if where and are extremal chains of type one and type two (their definitions are given in the main text), respectively. Thus we extend the known results of extremal benzenoid chains on Wiener number to a more general case.

Correlation analysis of the π-electronic energy and the second-order general connectivity index for benzenoid chains

Benzenoid chains are natural graph representations of chain hydrocarbons. For a given benzenoid chain [Formula: see text], we find an expression of its second-order general connectivity index, denoted by [Formula: see text], in terms of inlet features of [Formula: see text]. It is also shown that the second-order general connectivity index [Formula: see text] correlates well with the [Formula: see text]-electronic energy of [Formula: see text] for [Formula: see text], especially for [Formula: see text].

The edge-Hosoya polynomial of benzenoid chains

The Hosoya polynomial is a well known vertex-distance based polynomial, closely correlated to the Wiener index and the hyper-Wiener index, which are widely used molecular-structure descriptors. In the present paper we consider the edge version of the Hosoya polynomial. For a connected graph G let $$d_e(G,k)$$ be the number of (unordered) edge pairs at distance k. Then the edge-Hosoya polynomial of G is $$H_e(G,x) = \sum _{k \ge 0} d_e(G,k)\,x^k$$ . We investigate the edge-Hosoya polynomial of important chemical graphs known as benzenoid chains and derive the recurrence relations for them. These recurrences are then solved for linear benzenoid chains, which are also called polyacenes.

A study about the Tutte polynomials of benzenoid chains

Abstract The Tutte polynomials for signed graphs were introduced by Kauffman. In 2012, Fath-Tabar, Gholam-Rezaeı and Ashrafı presented a formula for computing Tutte polynomial of a benzenoid chain. From this point on, we have also calculated the Tutte polynomials of signed graphs of benzenoid chains in this study.