Let Pn be a pentagonal chain. Motivated by the work of Gutman (1977), this paper shows that for a hexagonal chain H, there exists a caterpillar tree T(H) such that the number of Kekulé structures of ...H is equal to the Hosoya index of T(H). In this paper, we show that for a pentagonal chain Pn with even number of pentagons, there exists a caterpillar tree Tn2 such that the number of Kekulé structures of Pn is equal to the Hosoya index of Tn2. This result can be generalized to any polygonal chain Qn with even number of odd polygons.
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.
Haruo Hosoya initiated Hosoya Index (Z Index) in 1971. It is used in chemical informatics for examining organic compounds. Hosoya introduced Z index denoted by H (G) to report a good correlation of ...the boiling points of alkane isomers and showed certain physico-chemical properties of saturated hydrocarbons 6. In this paper, the study of Hosoya index is extended to interconnection networks and Z index for triangular snake graph and alternate triangular snake graph using matching, Z counting polynomial and matching polynomial is obtained.
The Wiener polarity index W sub(p)(G) of a graph G is the number of unordered pairs of vertices {u, v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied ...the Wiener polarity index in Y. Zhang, Y. Hu, 2016 38. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus-Gaddum-type results for W sub(p)(G ). Our lower bound on View the MathML source Wp(G)+Wp(Gmacr) is always better than the previous lower bound given by Zhang and Hu.
Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm for that of the cycloparaphenylene (CPP) graphs in matrix ...product form. The algorithm gives a recursive relation in expressing the sum of the CP coefficients of a CPP in terms of that of its two immediately preceding analogues which ultimately ends up with the use of transfer matrix in an analytical form. The sum of CP coefficients, being combinatorial in nature, is found to be used as a topological index showing much similarity with Hosoya index (sum all matching polynomial coefficients), cardinality and number of Kekulé valence structures of CPP graphs compared to the Wiener index which is the distance sum of all pairs of vertices in the graph. The sum of CP coefficients has been found to model the physical properties like strain energy and diameter of CPPs that are verified by the respective excellent correlations.
Let Rn be a square–hexagonal chain. In this paper, we show that there exists a caterpillar tree Tn such that the number of Kekulé structures of Rn is equal to the Hosoya index of Tn. Since both ...hexagonal chains and polyomino chains can be viewed as special square–hexagonal chains, our result generalizes the corresponding results for hexagonal chains (Gutman, 1977) and polyomino chains (Liand Yan, 2012).
The polyphenyl system is composed of n hexagons, where consecutive hexagons are sticked by a path with two vertices. The Hosoya index of a graph G is defined as the total number of the independent ...edge sets of G. In this paper, we give two computation formulas of Hosoya index of two types of four leaves polyphenyl systems. In particular, we characterize the extremal Hosoya index of two types of four leaves polyphenyl systems.
A general approach to determine the matching polynomial (MP) of a graph with two parts connected by an edge is presented in matrix product that is ultimately used in deducing recursion formulas for ...obtaining the MP coefficients of linear and cylindrical poly(p-phenylene) (PPP) graphs. The Hosoya indices of linear and cylindrical PPPs are derived in terms of that of the two immediately preceding graphs as well as in analytical forms with the use of transfer matrices. Ambient condition density and bulk modulus of linear PPPs with 2-6 phenyl rings have been found to correlate well with the logarithm of their Hosoya indices. Excellent correlations of diameters with the logarithm of Hosoya indices and strain energies with the inverse of the logarithm of Hosoya indices for cylindrical PPPr with r (= 6-16, 18, 20) phenyl rings are obtained. The linear relation between the logarithm of Hosoya indices and diameter and the inverse relation between diameter and strain energy corroborate the fact.
The matching energy of a graph G is ME(G)=2π∫0∞1x2ln∑k≥0m(G,k)x2kdx, and the Hosoya index of G is Z(G)=∑k≥0m(G,k), where m(G,k) is the number of k-matchings in G. In this note, we first determine the ...maximum values of m(G,k) in all connected bipartite graphs with n vertices and a given connectivity. And then we determine the maximum matching energy (resp. Hosoya index) among all connected bipartite graphs with n vertices and a given (edge) connectivity and characterize the corresponding extremal graphs.
A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity ...connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2∈H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.
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