A positivity property of the dimer entropy of graphs

TL;DR

This paper investigates the positivity of the dimer entropy series coefficients in regular bipartite graphs, finding that violations decrease with larger graph size and conjecturing that positivity holds asymptotically.

Contribution

It introduces the concept of graph positivity for dimer entropy series and provides empirical evidence and conjectures about its prevalence in large regular bipartite graphs.

Findings

Violations occur mainly in small 3-regular bipartite graphs.

Frequency of violations decreases as graph size increases.

Positivity violations are rare in large graphs and in certain grid configurations.

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 ${\rm ln} N(i)/v$, 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 $4$.