1/n Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy

Using a 1/n expansion, that is an expansion in descending powers of n, for the number ofmatchings in regular graphs with 2n vertices, we study themonomer-dimer entropy for two classes of graphs. We study the difference between the extensive monomer-dimer entropy of a random r-regular graph G (bipartite or not) with 2n vertices and the average extensive entropy of r-regular graphs with 2n vertices, in the limit n -> infinity. We find a series expansion for it in the numbers of cycles; with probability 1 it converges for dimer density p < 1 and, for G bipartite, it diverges as | ln(1 - p)| for p -> 1. In the case of regular lattices, we similarly expand the difference between the specific monomer-dimer entropy on a lattice and the one on the Bethe lattice; we write down its Taylor expansion in powers of p through the order 10, expressed in terms of the number of totally reducible walks which are not tree-like. We prove through order 6 that its expansion coefficients in powers of p are non-negative.