On the Extremal Merrifield-Simmons Index of Quasi-Unicyclic Graphs

Abstract

The Merrifield-Simmons index \(i(G)\) of a graph \(G\) is defined as the total number of independent sets of \(G\). A connected graph \(G = (V,E)\) is called a quasi-unicyclic graph if there exists a vertex \(u_0 \in V\) such that \(G – u_0\) is a unicyclic graph. Denote by \(\mathcal{U}(n,d_0)\) the set of quasi-unicyclic graphs of order \(n\) with \(G – u_0\) being a unicyclic graph and \(d_G(u_0) = d_0\). In this paper, we characterize the quasi-unicyclic graphs with the smallest, the second-smallest, the largest, and the second-largest Merrifield-Simmons indices, respectively, in \(\mathcal{U}(n, d_0)\).