The Merrifield-Simmons Index and Hosoya Index of C ( n , k , λ ) Graphs

Abstract

The Merrifield-Simmons index $\mathrm{i(G)}$ of a graph $G$ is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets of $G$ The Hosoya index $\mathrm{z(G)}$ of a graph $G$ is defined as the total number of independent edge subsets, that is, the total number of its matchings. By $C(n,k,\lambda )$ we denote the set of graphs with $n$ vertices, $k$ cycles, the length of every cycle is $\lambda$, and all the edges not on the cycles are pendant edges which are attached to the same vertex. In this paper, we investigate the Merrifield-Simmons index $\mathrm{i(G)}$ and the Hosoya index $\mathrm{z(G)}$ for a graph $G$ in $C(n,k,\lambda )$.