TITLE: Maximum number of perfect matchings in a graph 
QUESTION [11 upvotes]: 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 $|E| = c|V|^2$.

REPLY [13 votes]: 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 1-factors" 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.