What is the maximal number of perfect matchings a graph $G(V,E)$ can have if $V$ and $E$ are fixed? I am particularly interested in a case when $E = cV^2$.
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I think this is exactly the main result of this recent paper we just published in Discrete Mathematics. Just in case the link doesn't work: this is "Graphs with the maximum or minimum number of 1factors" by D. Grossa, N. Kahl and J.T. Saccoman. I have read only the abstract. Let me know if this is what you were looking for. 

