Solving the dimer problem of the vertex-edge graph of a cubic graph

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $L(G)$ be the line graph of $G$, which has vertex set $E(G)$ and two vertices $e$ and $f$ of $L(G)$ is adjacent if $e$ and $f$ is incident in $G$. The vertex-edge graph $M(G)$ of $G$ has vertex set $V(G)\cup E(G)$ and edge set $E(L(G))\cup \{ue,ve|\forall e=uv\in E(G)\}$. In this paper, by a combinatorial technique, we show that if $G$ is a connected cubic graph with an even number of edges, then the number of dimer coverings of $M(G)$ equals $2^{|V(G)|/2+1}{3}^{|V(G)|/4}$. As an application, we obtain the exact solution of the dimer problem of the weighted solicate network obtained from the hexagonal lattice in the context of statistical physics.

13 pages, 4 figures

References in corpus (2)

• Exact solution of close-packed dimers on the kagome lattice
• Close-packed dimers on the kagome lattice: Finite lattices and the Grassmannian approach