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 $M_{G}(lambda)=sum_{k=0}^nm_k(G)lambda^k$‎. ‎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.