All Questions

Let $f(G)$ give the number of perfect matchings of a graph $G$. Consider set $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$. Consider collection of all $2n$ vertex balanced bipartite graph to be $\...

We have a $d$-regular bipartite graph $G = (X,Y,E)$ with $|X| = |Y| = n$ and $|E| = nd$. What is an upper bound on the number of perfect matchings of $G$?