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Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $L(G)$ bethe 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 combinatorialtechnique, we show that if $G$ is a connected cubic graph with an even numberof 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 solutionof the dimer problem of the weighted solicate network obtained from thehexagonal lattice in the context of statistical physics.

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