On uniqueness of the q-state Potts model on a self-dual family of graphs

TL;DR

This paper investigates the complex zeros of the Tutte polynomial for self-dual graphs, establishing regions with unique dominant eigenvalues that ensure analyticity of the pressure, and provides counterexamples to a known conjecture.

Contribution

It identifies specific regions where the eigenvalues are unique for self-dual graphs and presents counterexamples to a conjecture about the Tutte polynomial.

Findings

Regions with a single dominant eigenvalue are identified for certain self-dual graphs.

Counterexamples to Chen et al.'s conjecture are provided.

Examples include strips of triangles, wheels, and cycles with high multiplicity edges.

Abstract

This paper deals with the location of the complex zeros of the Tutte polynomial for a class of self-dual graphs. For this class of graphs, as the form of the eigenvalues is known, the regions of the complex plane can be focused on the sets where there is only one dominant eigenvalue in particular containing the positive half plane. Thus, in these regions, the analyticity of the pressure can be derived easily. Next, some examples of graphs with their Tutte polynomial having a few number of eigenvalues are given. The cases of the strip of triangles with a double edge, the wheel and the cycle with an edge having a high order of multiplicity are presented. In particular, for this last example, we remark that the well known conjecture of Chen et al. is false in the finite case.