Matchings in Benjamini–Schramm convergent graph sequences

Abstract

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree.A graph parameter is said to be estimable if it converges along every Benjamini-Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d-regular bipartite graphs for any fixed d ≥ 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge.However, for any sequence of d-regular bipartite graphs converging to the dregular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d-regular bipartite graphs. We show that the limit equals to the exponent in Schrijver's lower bound on the number of perfect matchings.Our analytic approach also yields a short proof for the Nguyen-Onak (also Elek-Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.

Matchings in vertex-transitive bipartite graphs

“…Note that Schrijver and Valiant proved in [18] that the number (d − 1) d−1 d d−2 cannot be improved by showing that for a random d-regular bipartite multigraph the statement is asymptotically tight. In [1] the authors proved that actually large girth graphs (not only random graphs) have asymptotically the same number of perfect matchings: let g(H) denote the girth of a graph H, i. e., the length of the shortest cycle in H. Then the following is true. Theorem 1.2 ([1]).…”

“…So in this case we do not need the vertex-transitivity of the graphs. On the other hand, in [1] the authors gave a sequence of d-regular bipartite graphs which are Benjamini-Schramm convergent, still the…”

“…We also noted in the Introduction that there are two inconvenient things in this statement. Namely, the term o v(G) (1), and that p is defined only for special values. It turns out that the two problems are in fact one.…”

“…Given a finite bipartite d-regular graph G and an edge e of G, let p(e) be the probability that a uniform random perfect matching contains e. The following theorem was proved in [1]. The consequence of this theorem was that Theorem 1.8 is not true without vertex-transitivity.…”

Chromatic roots and limits of dense graphs

“…∆. 1 Here, we suggest to bound the moduli of the chromatic roots by the order instead of the maximum degree.…”