The number of matchings in random graphs

TL;DR

This paper applies the cavity method to analyze matchings in sparse random graphs, providing new entropy calculations for matchings of given sizes and validating known results for maximum and perfect matchings.

Contribution

It introduces an algorithm for computing the entropy of matchings in large graphs with diverging girth and derives analytic entropy formulas for regular and Erdos-Renyi random graphs.

Findings

Reproduces known results for maximum and perfect matchings.

Provides a new algorithm for entropy computation in large graphs.

Derives analytic entropy formulas for specific random graph ensembles.

Abstract

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.