Solving combinatorially the monomer-dimer problem on certain fractal scale-free lattices

Counting monomer-dimers, or equivalently, matchings of a graph G, is an interesting but intriguing question with great difficulties in statistical physics. Let Z(G) denote the number of monomer-dimers of G. In this paper, we propose a new combinatorial method to solve this problem in some special kind of graphs R(G;D;Haibi), which are obtained from a graph G with edge set E by replacing each edge e=uivi∈E﹨D with a graph Haibi having two root vertices ai and bi by identifying ui with ai and vi with bi for i=1,2,…,|E|−|D|, where D is a matching of G, and Haibi satisfies: Z(Haibi)Z(Haibi−ai−bi)=Z(Haibi−ai)Z(Haibi−bi). As an application, we obtain the exact solution of the monomer-diner problem on a fractal scale-free lattice, which answers a question posed by Zhang et al. in 2011.