On the monomer-dimer problem of some graphs

摘要

The pure-dimer problem was solved in exact closed form for many lattice graphs. Although some numerical solutions of the monomer-dimer problem were obtained, no exact solutions of the monomer-dimer problem were available (except in one dimension). Let G be an arbitrary graph with N vertices. Construct a new graph R(G) from G by adding a new verex e* corresponding to each edge e = (a, b) of G and by joining each new vertex e* to the vertices a and b. If the suitable activities of vertices and edges in R(G) are selected, then the monomer-dimer problem can be solved exactly for the graph R(G), which generalizes the result obtained by Yan and Yeh. As applications, if we select suitable activities for the vertices and edges of R((L) over tilde (n,d)), we obtain the exact formulae for the MD partition function, MD free energy, and MD entropy of R((L) over tilde (n,d)) for the d-dimensional lattice (L) over tilde (n,d)) with periodic boundaries.