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 $M_{G}(\lambda)=\sum_{k=۰}^nm_k(G)\lambda^k$‎. ‎In this paper we show that if $G$ is a $d$--regular graph and $۰<\lambda<(۴d)^{-۲}$‎, ‎then‎ ‎$\frac{۱}{v(G)}\ln M_G(\lambda)>\frac{۱}{v(K_{d+۱})}\ln M_{K_{d+۱}}(\lambda).$‎ ‎The same inequality holds true if $d=۳$ and $\lambda<۰.۳۵۷۵$‎. ‎More precise conjectures are also given‎.

کلیدواژه ها:

‎matchings‎ ، ‎matching polynomial‎ ، ‎regular graphs