University of Isfahan Transactions on Combinatorics 2251-8657 10 2 2021 06 01 Matchings in regular graphs‎: ‎minimizing the partition function 73 95 25093 10.22108/toc.2020.123763.1742 EN Márton Borbényi Eötvös Loránd University, Budapest, Hungary Peter Csikvari Eötvös Loránd University, Budapest, Hungary Journal Article 2020 07 01 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 $0<\lambda<(4d)^{-2}$‎, ‎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. https://toc.ui.ac.ir/article_25093_0b0489f0c8b0daaa064de163958fb5a0.pdf