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.
Note that our theorem gives a generating function of the moments of @math since for @math sufficiently small, we have: z ^2 1+z ^2 d T(G) ( ) = i=1 ^ (-1)^ i+1 z^i ^ 2i d T(G) ( ), and the series is convergent since the support of all the @math and hence of @math is contained in @math . As shown by Godsil in @cite_9 , for finite trees, the spectral measure and the matching measure coincide, this is still true for infinite trees @cite_18 @cite_23 @cite_38 . In particular, the moments of the matching measure @math can be interpreted as the average number of closed walks on @math where the average is taken over the starting point of the walk (see Proposition for a precise definition of the random root as the starting point of the walk). To be more precise, for a finite graph @math , we denote by @math the real eigenvalues of its adjacency matrix and we define the empirical spectral measure of the graph @math as the probability measure on @math : = 1 v(G) i=1 ^ v(G) .
@cite_38 @cite_9 @cite_18 @cite_23
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The matching polynomial α(G, x) of a graph G is a form of the generating function for the number of sets of k independent edges of G. in this paper we show that if G is a graph with vertex v then there is a tree T with vertex w such that This result has a number of consequences. Here we use it to prove that α(G , 1 x) xα(G, 1 x) is the generating function for a certain class of walks in G. As an application of these results we then establish some new properties of α(G, x). We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieltjes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and graphs with a given degree sequence. We give examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010 We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs @math converging locally to a Galton--Watson tree @math (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function @math of @math . In the first part, we show that the adjacency operator associated with @math is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on @math for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of @math . Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.