Exact Potts/Tutte Polynomials for Hammock Chain Graphs

TL;DR

This paper provides exact calculations of Potts model partition functions and Tutte polynomials for complex chain graphs made of hammock subgraphs, analyzing their properties and zero loci under different boundary conditions.

Contribution

It introduces a method to compute exact Potts and Tutte polynomials for hammock chain graphs with various boundary conditions, including analysis of their zero loci.

Findings

Exact formulas for Potts and Tutte polynomials for hammock chain graphs.

Analysis of the zero loci of the Potts partition function in the complex plane.

Special cases include chromatic, flow, and reliability polynomials.

Abstract

We present exact calculations of the $q$-state Potts model partition functions and the equivalent Tutte polynomials for chain graphs comprised of $m$ repeated hammock subgraphs $H_{e_1,...,e_r}$ connected with line graphs of length $e_g$ edges, such that the chains have open or cyclic boundary conditions (BC). Here, $H_{e_1,...,e_r}$ is a hammock (series-parallel) subgraph with $r$ separate paths along ``ropes'' with respective lengths $e_1, ..., e_r$ edges, connecting the two end vertices. We denote the resultant chain graph as $G_{\{e_1,...,e_r\},e_g,m;BC}$. We discuss special cases, including chromatic, flow, and reliability polynomials. In the case of cyclic boundary conditions, the zeros of the Potts partition function in the complex $q$ function accumulate, in the limit $m \to \infty$, onto curves forming a locus ${\cal B}$, and we study this locus.