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{\bf S. Friedland, E. Krop and K. Markstr\"om}
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{\bf On the Number of Matchings in Regular Graphs}
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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 $m\le 4$.

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