We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees.
Recall that @math is called the partition function. A crucial observation made by Csikv 'ari @cite_48 is that for any @math -lift @math of a bipartite graph @math , we have @math , see Proposition in the sequel. Note that a @math -lift of a @math -regular graph is still @math -regular. Starting form any @math -regular bipartite graph @math , Csikv 'ari builds a sequence of @math -lifts with increasing girth. Then Csikv 'ari uses the framework of local weak convergence in order to define a limiting graph for the sequence of @math -lifts. In this particular case, the limit is the infinite @math -regular tree. Using the connection between the local weak convergence and the matching measure of a graph developed in @cite_31 , Csikv 'ari computes a limiting partition function associated to this infinite @math -regular tree which is obtained from the Kesten-McKay measure. This limiting partition function is then a lower bound for the original partition function @math and the lower bounds ) and ) are then easily obtained by properly choosing the parameter @math as a function of @math the size of the matchings that we need to count.
@cite_48 @cite_31
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Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs. We define the matching measure of a lattice L as the spectral measure of the tree of self-avoiding walks in L. We connect this invariant to the monomer–dimer partition function of a sequence of finite graphs converging to L. This allows us to express the monomer–dimer free energy of L in terms of the matching measure. Exploiting an analytic advantage of the matching measure over the Mayer series then leads to new, rigorous bounds on the monomer–dimer free energies of various Euclidean lattices. While our estimates use only the computational data given in previous papers, they improve the known bounds significantly.