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 $S_n^{l,k}\ast$ 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 $S_n^{l,k}\ast$ 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 $k=n–3$, $S_n^{3,k}$ has the maximum Merrifield-Simmons index among all graphs in $G(n,k)$; When $k=1,n-4$, $S_n^{4,k}$ or $S_n^{n-k,k}$ has the maximum Merrifield-Simmons index among all graphs in $G(n,k)$
2. When $2\le k\le n-5$, $S_n^{n-k,k}$ and $S_n^{4,k}$ 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.