On the maximum number of maximum independent sets in connected graphs

We characterize the connected graphs of given order n and given independence number α that maximize the number of maximum independent sets. For 3 ≤ α ≤ n ∕ 2 , there is a unique such graph that arises from the disjoint union of α cliques of orders ⌈ n α ⌉ and ⌊ n α ⌋ , which is the complement of a Turán graph, by selecting a vertex x in a largest clique and adding an edge between x and a vertex in each of the remaining α − 1 cliques. Our result confirms a conjecture of Derikvand and Oboudi.