Let $G$ be a simple graph. An independent set is a set of pairwise non-adjacent vertices. The number of vertices in a maximum independent set of $G$ is denoted by $\alpha(G)$. In this paper, we characterize graphs $G$ with $n$ vertices and with maximum number of maximum independent sets provided that $\alpha(G)\leq 2$ or $\alpha(G)\geq n-3$.

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