Friedland's Lower Matching Conjecture asserts that if G is a d--regular
bipartite graph on v(G)=2n vertices, and mk(G) denotes the number of
matchings of size k, then mk(G)≥(kn)2(dd−p)n(d−p)(dp)np, where p=nk. When
p=1, this conjecture reduces to a theorem of Schrijver which says that a
d--regular bipartite graph on v(G)=2n vertices has at least
(dd−2(d−1)d−1)n perfect matchings. L. Gurvits
proved an asymptotic version of the Lower Matching Conjecture, namely he proved
that v(G)lnmk(G)≥21(pln(pd)+(d−p)ln(1−dp)−2(1−p)ln(1−p))+ov(G)(1).
In this paper, we prove the Lower Matching Conjecture. In fact, we will prove
a slightly stronger statement which gives an extra cpn factor
compared to the conjecture if p is separated away from 0 and 1, and is
tight up to a constant factor if p is separated away from 1. We will also
give a new proof of Gurvits's and Schrijver's theorems, and we extend these
theorems to (a,b)--biregular bipartite graphs