Matchings in regular graphs‎: ‎minimizing the partition function

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\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.

Keywords

• ‎matchings‎
• ‎matching polynomial‎
• ‎regular graphs