The number of independent sets of unicyclic graphs with given matching number

Abstract

The Hosoya index $z\left(G\right)$ of a graph $G$ is defined as the number of matchings of $G$ and the Merrifield–Simmons index $i\left(G\right)$ of $G$ is defined as the number of independent sets of $G$. Let $U(n,m)$ be the set of all unicyclic graphs on $n$ vertices with ${\alpha }^{\prime }\left(G\right)=m$. Denote by $U^1(n,m)$ the graph on $n$ vertices obtained from $C_3$ by attaching $n-2m+1$ pendant edges and $m-2$ paths of length 2 at one vertex of $C_3$. Let $U^2(n,m)$ denote the $n$-vertex graph obtained from $C_3$ by attaching $n-2m+1$ pendant edges and $m-3$ paths of length 2 at one vertex of $C_3$, and one pendant edge at each of the other two vertices of $C_3$. In this paper, we show that $U^1(n,m)$ and $U^2(n,m)$ have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield–Simmons index among all graphs in $U(n,m)\setminus \left\{C_n\right\}$, respectively.