Let G be a simple graph with vertex set and edge set . The function which assigns to each pair of vertices in , the length of minimal path from to , is called the distance function between two vertices. The distance function between and edge and a vertex is where for and. , . The Wiener index of a graph is denoted by and is defined by .In general this kind of index is called a topological index...

A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based indices such as Wiener index, hyper-Wiener Harary Tratch – Stankevitch Zefirov index. In present study, we determine closed form for third type chain hex-derived network dime...

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...

Abstract The n-th order Wiener index of a molecular graph G was put forward by Estrada et al. [New J. Chem. 22 (1998) 819] as ( ) 1 ( , ) n n x W H G x where ( , ) H G x is the Hosoya polynomial. Recently Brückler et al. [Chem. Phys. Lett. 503 (2011) 336] considered a related graph invariant, ( ) 1 1 (1/ !) ( ( , )) / n n n n x W n d x H G x d x . For n=1, both W and W reduce to the ordinary W...

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di σi denote, respectively, degree transmission of vi, for 1≤i≤n. In this paper, we aim to provide new matrix description celebrated Wiener index. fact, introduce Wiener–Hosoya G, which is defined as n×n whose (i,j)-entry equal σi2di+σj2dj if vi vj are adjacent 0 otherwise. Some properties, including upper lower ...

An induced subgraph H of a graph G is gated if for every vertex x outside H there exists a vertex x ′ inside H such that each vertex y of H is connected with x by a shortest path passing through x . The gated amalgam of graphs G1 and G2 is obtained from G1 and G2 by identifying their isomorphic gated subgraphs H1 and H2. Two theorems on Hosoya polynomials of gated amalgams are provided. As thei...

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method...