Counting matchings in irregular bipartite graphs and random lifts

TL;DR

This paper establishes sharp lower bounds on the number of matchings in bipartite graphs, extending to regular, bi-regular, and random lifts, and connects these bounds to permanents and spectral measures.

Contribution

It provides new lower bounds for matchings and permanents in bipartite graphs, extending existing conjectures and theorems, and introduces an algorithmic proof approach.

Findings

Lower bounds are order optimal and attained for graph lifts.

Results recover and extend Friedland's and Schrijver's theorems.

New bounds for subpermanents and insights into spectral measures of lifts.

Abstract

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 $2$-lifts of the original graph as well as for random $n$-lifts of the original graph when $n$ 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 $n$-lifts with some implications for the matching measure and the spectral measure of random $n$-lifts as well as for the spectral measure of infinite trees.