End Graph Effects on Chromatic Polynomials for Strip Graphs of Lattices and their Asymptotic Limits

TL;DR

This paper calculates the chromatic polynomials and their asymptotic limits for strip graphs of lattices, revealing complex nonanalytic regions in the complex plane and exploring effects of end graph configurations.

Contribution

It provides exact calculations of ground state degeneracy for Potts antiferromagnets on strip graphs with various end conditions, extending analysis to complex q values and identifying new nonanalytic regions.

Findings

Enclosed nonanalytic regions in the complex q plane for planar end graphs.

First example of such regions for this class of strip graphs.

Analysis focused on a lattice segment of the $(3^3 b7 4^2)$ Archimedean lattice.

Abstract

We report exact calculations of the ground state degeneracy per site (exponent of the ground state entropy) $W(\{G\},q)$ of the $q$-state Potts antiferromagnet on infinitely long strips with specified end graphs for free boundary conditions in the longitudinal direction and free and periodic boundary conditions in the transverse direction. This is equivalent to calculating the chromatic polynomials and their asymptotic limits for these graphs. Making the generalization from $q \in {\mathbb Z}_+$ to $q \in {\mathbb C}$, we determine the full locus ${\cal B}$ on which $W$ is nonanalytic in the complex $q$ plane. We report the first example for this class of strip graphs in which ${\cal B}$ encloses regions even for planar end graphs. The bulk of the specific strip graph that exhibits this property is a part of the $(3^3 \cdot 4^2)$ Archimedean lattice.