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 $M_G(\lambda )={\sum }_{k=0}^n{m}_k(G){\lambda }^k$. 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.