Extremal Multi-bridge Graphs With Respect To Merrifield-Simmons Index

Abstract

The Merrifield-Simmons index of a graph $G$, denoted by $i(G)$, is defined to be the total number of its independent sets, including the empty set. Let $\theta (a_1,a_2,\dots ,a_k)$ denote the graph obtained by connecting two distinct vertices with $k$ independent paths of lengths $a_1,a_2,\dots ,a_k$ respectively, we named it as multi-bridge graphs for convenience. Tight upper and lower bounds for the Merrifield-Simmons index of $\theta (a_1,a_2,\dots ,a_k)$ are established in this paper.