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

Shubo Chen1, Weijun Liu2
1 Department of Mathematics and Computer Science, Hunan City University, Yiyang, Hunan 413000, P. R. China
2 School of Sciences, Nantong University, Nantong, Jiangsu, 226007, P. R. China

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 \(\theta(a_1, a_2, \ldots, a_k)\) denote the graph obtained by connecting two distinct vertices with \(k\) independent paths of lengths \(a_1, a_2, \ldots, a_k\) respectively, we named it as multi-bridge graphs for convenience. Tight upper and lower bounds for the Merrifield-Simmons index of \(\theta(a_1, a_2, \ldots, a_k)\) are established in this paper.