Matchings in regular graphs: minimizing the partition function

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 $M_{G}(λ)=\sum_{k=0}^nm_k(G)λ^k$. In this paper we show that if $G$ is a $d$--regular graph and $0<λ<(4d)^{-2}$, then $\frac{1}{v(G)}\ln M_G(λ)>\frac{1}{v(K_{d+1})}\ln M_{K_{d+1}}(λ).$ The same inequality holds true if $d=3$ and $λ<0.3575$. More precise conjectures are also given.

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