CONNECTED GRAPHS WITH A LARGE NUMBER OF INDEPENDENT SETS

Abstract

For a simple undirected graph $G=(V(G),E(G))$, a subset $I$ of $V(G)$ is said to be an independent set of $G$ if any two vertices in $I$ are not adjacent in $G$. An empty set is also an independent set in $G$. The set of all independent sets of a graph $G$ is denoted by $I(G)$ and its cardinality by $i(G)$ (known as the Merrifield-Simmons index in mathematical chemistry). Let $h(n,x)$ be the $x$-th largest number of independent sets among all connected $n$-vertex graphs. In this paper, we determine the numbers $h(n,x)$ for $1\le x\le \lfloor\frac{n}{2}\rfloor ^2-3\cdot\lfloor\frac{n}{2}\rfloor+3$. Besides, we also characterize the connected $n$-vertex graphs achieving these values.