Simple expressions for estimating the molecular freedom parameter in the dimer problem

A method for calculating the molecular freedom parameter in the dimer model for two-parameter families of lattice graphs with periodic boundary conditions is discussed. It is shown that the scheme of calculations for the most known types of lattices can be unified and it allows us to find simple but fairly accurate estimates from above of the model parameter. The possibilities of the proposed approach are illustrated by examples for which only numerical values were previously known. It is noted that the method can also be useful for estimating parameters in other lattice models. We show that the entropy of spanning trees on a triangular lattice is a special case of the formulas derived in this paper.