14-63 P. Butera, P. Federbush, M. Pernici
Dimer entropy of a graph and a positivity property (52K, LaTeX) Aug 3, 14
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Abstract. 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 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 r-regular bipartite graphs, goes to zero as v tends to infinity.

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