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 ...

Graphical matrices are presented. Their construction via selected sets of subgraphs and the replacement of subgraphs by numbers representing graph invariants are discussed. The last step of the procedure is to apply the method of choice for obtaining the desired double invariant from the graphical matrix in the numerical form. It is also pointed out that many so-called special graph-theoretical...

In this paper, Kekule structures of benzenoid chains are considered. It has been shown that the coefficients of a B_n (x) Morgan-Voyce polynomial equal to the number of k-matchings (m(G,k)) of a path graph which has N=2n+1 points. Furtermore, two relations are obtained between regularly zig-zag nonbranched catacondensed benzenid chains and Morgan-Voyce polynomials and between regularly zig-zag ...

let $g=(v,e)$ be a‎ ‎simple graph of order $n$ and size $m$‎. ‎an $r$-matching of $g$ is ‎a set of $r$ edges of $g$ which no two of them have common vertex‎. ‎the hosoya index $z(g)$ of a graph $g$ is defined as the total‎ ‎number of its matchings‎. ‎an independent set of $g$ is a set of‎ ‎vertices where no two vertices are adjacent‎. ‎the ‎merrifield-simmons index of $g$ is defined as the tota...