Matchings in regular graphs‎: ‎minimizing the partition function

Document Type : Research Paper


Eötvös Loránd University, Budapest, Hungary


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.


Main Subjects

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  • Receive Date: 01 July 2020
  • Revise Date: 01 November 2020
  • Accept Date: 01 November 2020
  • Published Online: 01 June 2021