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 $|E| = c|V|^2$. 



Maximum number of perfect matchings in a graph 
I think this is exactly the main result of [this recent paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V00-4X9D5FW-2&_user=4423&_coverDate=02%2F28%2F2010&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1298182872&_rerunOrigin=scholar.google&_acct=C000059605&_version=1&_urlVersion=0&_userid=4423&md5=e0f086e2b1298619e290e92b111dc6b8) 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. 


