 1463 P. Butera, P. Federbush, M. Pernici
 Dimer entropy of a graph and a positivity property
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Aug 3, 14

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Abstract. The entropy of a monomerdimer system on an infinite bipartite
lattice can be written as a meanfield 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 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 count of the dimer configuration 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, all the non
positive graphs have vertices of degree 3. From v=14 to v=28,
and degree 3, a few violations of the positivity occur, but their
frequency decreases with increasing v. We conjecture that
for each degree r the frequency of violations, in the class of the
rregular bipartite graphs, goes to zero as v tends to
infinity.
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