  We show that the ratio of the number of near perfect matchings to the number
of perfect matchings in $d$-regular strong expander (non-bipartite) graphs,
with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov
chain [JS89] mixes in polynomial time and generates an (almost) uniformly
random perfect matching. Furthermore, we prove that such graphs have at least
$\Omega(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture
[LP86] for this family of graphs.
