  For the set of graphs with a given degree sequence, consisting of any number
of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the
optimal graphs who maximize and minimize the number of $m$-matchings. We find
the expected value of the number of $m$-matchings of $r$-regular bipartite
graphs on $2n$ vertices with respect to the two standard measures. We state and
discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular
bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite
$r$-regular bipartite graphs. We prove these conjectures for 2-regular
bipartite graphs and for $m$-matchings with $m\le 4$.
