Summary: An independent set in a graph \(G\) is a set of pairwise nonadjacent vertices of \(G\). The independence number, \(\alpha\), of \(G\) is the maximum cardinality of an independent set in \(G\). An independent set in \(G\) is maximum if it has cardinality \(\alpha\). E. Mohr and D. Rautenbach [Graphs Comb. 34, No. 6, 1729–1740 (2018; Zbl 1402.05169)] determined the \(n\)-vertex trees (resp. connected graphs, disconnected graphs) with independence number \(\alpha\) having the largest number of maximum independent sets. As a continuance of these works, we give complete characterizations among the following families of graphs:

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the \(n\)-vertex forests with independence number \(\alpha\) having the first three largest number of maximum independent sets;

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the \(n\)-vertex trees with independence number \(\alpha\) having the second and third largest number of maximum independent sets;

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the bipartite graphs (containing at least one cycle) of order \(n\) and independence number \(\alpha\) having the maximum number of maximum independent sets;

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the disconnected \(n\)-vertex graphs with independence number \(\alpha\) having the second largest number of maximum independent sets.

Furthermore, we obtain a complete classification of connected graphs with order \(n\) and independence number \(n - 3\), which gives a solution to an open problem of Derikvand and Oboudi.