Sharp Bounds on Merrifield-Simmons Index of the Generalized $\theta$-Graph

Abstract

For a graph $G$, the Merrifield-Simmons index $i(G)$ is defined as the total number of its independent sets. In this paper, we determine sharp upper and lower bounds on Merrifield-Simmons index of generalized $\theta$-graph, which is obtained by subdividing the edges of the multigraph consisting of $k$ parallel edges, denoted by $\theta (r_1,r_2,\dots ,r_k)$. The corresponding extremal graphs are also characterized.