  The entropy of a monomer-dimer system on an infinite bipartite lattice can be
written as a mean-field part plus a series expansion in the dimer density. In a
previous paper it has been conjectured that all coefficients of this series are
positive. Analogously on a connected regular graph with $v$ vertices, the
"entropy" of the graph ${\rm ln} N(i)/v$, where $N(i)$ is the number of ways of
setting down $i$ dimers on the graph, can be written as a part depending only
on the number of the dimer configurations over the completed graph plus a
Newton series in the dimer density on the graph. In this paper, we investigate
for which connected regular graphs all the coefficients of the Newton series
are positive (for short, these graphs will be called positive). In the class of
connected regular bipartite graphs, up to $v=20$, the only non positive graphs
have vertices of degree $3$. From $v=14$ to $v=30$, the frequency of the
positivity violations in the $3$-regular graphs decreases with increasing $v$.
In the case of connected $4$-regular bipartite graphs, the first violations
occur in two out of the $2806490$ graphs with $v=22$. We conjecture that for
each degree $r$ the frequency of the violations, in the class of the
$r-$regular bipartite graphs, goes to zero as $v$ tends to infinity. This
graph-positivity property can be extended to non-regular or non-bipartite
graphs. We have examined a large number of rectangular grids of size $N_x
\times N_y $ both with open and periodic boundary conditions. We have observed
positivity violations only for $min(N_x, N_y) = 3$ or $4$.

High Energy Physics - Lattice;Statistical Mechanics