On Maximum Merrifield-Simmons Index of Unicyclic Graphs with Prescribed Pendent Vertices

Abstract

The Merrifield-Simmons index \(\sigma(G)\) of a (molecular) graph \(G\) is defined as the number of independent-vertex sets of \(G\). By \(G(n, l, k)\) we denote the set of unicyclic graphs with girth \(l\) and the number of pendent vertices being \(k\) respectively. Let \(S_n^l\) be the graph obtained by identifying the center of the star \(S_{n-l+1}\) with any vertex of \(C_l\). By \(S^{l,k}_n*\) we denote the graph obtained by identifying one pendent vertex of the path \(P_{n-l-k+1}\) with one pendent vertex of \(S_{l+k}^l\). In this paper, we first investigate the Merrifield-Simmons index for all unicyclic graphs in \(G(n,l,k)\) and \(S^{l,k}_n*\) is shown to be the unique unicyclic graph with maximum Merrifield-Simmons index among all unicyclic graphs in \(G(n, l, k)\) for fixed \(l\) and \(k\). Moreover, we proved that:

1. When \(k = n – 3\), \(S^{3,k}_n\) has the maximum Merrifield-Simmons index among all graphs in \(G(n, k)\); When \(k = 1, n-4\), \(S^{4,k}_n\) or \(S^{n-k,k}_n\) has the maximum Merrifield-Simmons index among all graphs in \(G(n,k)\)
2. When \(2 \leq k \leq n-5\), \(S^{n-k,k}_n\) and \(S^{4,k}_n\) are respectively unicyclic graphs having maximum and second-maximum Merrifield-Simmons indices among all unicyclic graphs in \(G(n, k)\), where \(G(n, k)\) denotes the set of unicyclic graphs with \(n\) vertices and \(k\) pendent vertices.