On 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 V(L(G))=E(G) and two vertices e and f in L(G) are adjacent if and only if two edges e and f in G have a common vertex. The vertex-edge graph M(G) of G has vertex set V(G)∪E(G) and edge set E(L(G))∪{ue,ve|∀e=uv∈E(G)}. In this paper, 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+13|V(G)|/4. As an application, we obtain the exact solution of the dimer problem of the weighted silicate network obtained from the hexagonal lattice in the context of statistical physics.