Matchings in regular graphs: minimizing the partition function

TL;DR

This paper investigates the behavior of the partition function related to matchings in regular graphs, establishing inequalities that compare these functions across different graph structures within certain parameter ranges.

Contribution

It proves new lower bounds for the normalized logarithm of the matching partition function in regular graphs, extending known inequalities to specific cases and parameter ranges.

Findings

For $d$-regular graphs, the inequality holds when $0<\lambda<(4d)^{-2}$.

The inequality also holds for $d=3$ when $\lambda<0.3575$.

Conjectures for more general cases are proposed.

Abstract

For a graph $G$ on $v(G)$ vertices let $m_k(G)$ denote the number of matchings of size $k$, and consider the partition function $M_{G}(\lambda)=\sum_{k=0}^nm_k(G)\lambda^k$. In this paper we show that if $G$ is a $d$--regular graph and $0<\lambda<(4d)^{-2}$, then $$\frac{1}{v(G)}\ln M_G(\lambda)>\frac{1}{v(K_{d+1})}\ln M_{K_{d+1}}(\lambda).$$ The same inequality holds true if $d=3$ and $\lambda<0.3575$. More precise conjectures are also given.