THE SECOND LARGEST NUMBER OF MAXIMAL INDEPENDENT SETS IN GRAPHS WITH AT MOST k CYCLES

Abstract

Let $G$ be a simple undirected graph. Denote by $\mbox{ mi}(G)$ (respectively, $\mbox{xi}(G)$) the number of maximal (respectively, maximum) independent sets in $G$. In this paper we determine the second largest value of $\mbox{mi}(G)$ for graphs with at most $k$ cycles. Extremal graphs achieving these values are also determined.