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Using a relation between the virial expansion coefficients of the pressureand the entropy expansion coefficients in the case of the monomer-dimer modelon infinite regular lattices, we have shown that, on hypercubic lattices of anydimension, the virial coefficients are positive through the 20th order. We haveobserved that all virial coefficients so far known for this system are positivealso on infinite regular lattices with different structure. We are thus led toconjecture that the virial expansion coefficients $m_k $ are always positive.These considerations can be extended to the study of related bounds on finitegraphs generalizing the infinite regular lattices, namely the finite grids andthe regular biconnected graphs. The validity of the bounds $\Delta^k {\rmln}(i! N(i)) \le 0$ for $k \ge 2$, where $N(i)$ is the number of configurationsof $i$ dimers on the graph and $\Delta$ is the forward difference operator, isshown to correspond to the positivity of the virial coefficients. Our tests onmany finite lattice graphs indicate that on large lattices these bounds aresatisfied, giving support to the conjecture on the positivity of the virialcoefficients. An exhaustive survey of some classes of regular biconnectedgraphs with a not too large number $v$ of vertices shows only few violations ofthese 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 graphsand for regular graphs. The similarity between the Heilman-Lieb inequality andthe 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 knowncoefficients of the infinite regular lattice models.

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