The Extremal Fully Loaded Graphs with Respect to Merrifield-Simmons Index

Abstract

The Merrifield-Simmons index, denoted by \(i(G)\), of a graph \(G\) is defined as the total number of its independent sets. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than \(3\) in its unique cycle. Let \(\mathcal{U}_n^1\) be the set of fully loaded unicyclic graphs. In this paper, we determine graphs with the largest, second-largest, and third-largest Merrifield-Simmons index in \(\mathcal{U}_n^1\).