  A theorem of A. Schrijver asserts that a $d$-regular bipartite graph on $2n$
vertices has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$ perfect
matchings. L. Gurvits gave an extension of Schrijver's theorem for matchings of
density $p$. In this paper we give a stronger version of Gurvits's theorem in
the case of vertex-transitive bipartite graphs. This stronger version in
particular implies that for every positive integer $k$, there exists a positive
constant $c(k)$ such that if a $d$-regular vertex-transitive bipartite graph on
$2n$ vertices contains a cycle of length at most $k$, then it has at least
$$\left(\frac{(d-1)^{d-1}}{d^{d-2}}+c(k)\right)^n$$ perfect matchings.
  We also show that if $(G_i)$ is a Benjamini--Schramm convergent graph
sequence of vertex-transitive bipartite graphs, then $$\frac{\ln
pm(G_i)}{v(G_i)}$$ is convergent, where $pm(G)$ and $v(G)$ denote the number of
perfect matchings and the number of vertices of $G$, respectively.
  We also show that if $G$ is $d$-regular vertex-transitive bipartite graph on
$2n$ vertices and $m_k(G)$ denote the number of matchings of size $k$, and
$$M(G,t)=1+m_1(G)t+m_2(G)t^2+\dots +m_n(G)t^n=\prod_{k=1}^n(1+\gamma_k(G)t),$$
where $\gamma_1(G)\leq \dots \leq \gamma_n(G)$, then $$\gamma_k(G)\geq
\frac{d^2}{4(d-1)}\frac{k^2}{n^2},$$ and $$\frac{m_{n-1}(G)}{m_n(G)}\leq
\frac{2}{d}n^2.$$ The latter result improves on a previous bound of C. Kenyon,
D. Randall and A. Sinclair. There are examples of $d$-regular bipartite graphs
for which these statements fail to be true without the condition of
vertex-transitivity.
