Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Using an inequality by Heilmann and Lieb, we find rigorous upper bounds on N(i) valid for arbitrary graphs and for regular graphs. The similarity between this inequality and the one conjectured above suggests that one study the stricter inequality mk≥12k for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.