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

Using a relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer-dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients $m_k $ are always positive. These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds $\Delta^k {\rm ln}(i! N(i)) \le 0$ for $k \ge 2$, where $N(i)$ is the number of configurations of $i$ dimers on the graph and $\Delta$ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients. Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. An exhaustive survey of some classes of regular biconnected graphs with a not too large number $v$ of vertices shows only few violations of these bounds. We conjecture that the frequency of the violations vanishes as $v \to \infty$. We find rigorous upper bounds on $N(i)$ valid for arbitrary graphs and for regular graphs. The similarity between the Heilman-Lieb inequality and the one conjectured above suggests that one study the stricter inequality $m_k \ge \frac{1}{2k}$ for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.