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

TL;DR

This paper derives an exact formula for the number of dimer coverings on the vertex-edge graph of a connected cubic graph with an even number of edges, linking combinatorics and statistical physics.

Contribution

It introduces a combinatorial method to solve the dimer problem on the vertex-edge graph of cubic graphs and applies it to a physical lattice model.

Findings

Number of dimer coverings equals 2^{|V(G)|/2+1}3^{|V(G)|/4} for the specified graphs.

Provides an exact solution for the dimer problem on a weighted solicate network derived from the hexagonal lattice.

Connects combinatorial graph theory with statistical physics models.

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.