Chromatic Polynomials for $J(\prod H)I$ Strip Graphs and their Asymptotic Limits

TL;DR

This paper computes chromatic polynomials for complex strip graphs with various boundary conditions, analyzes their asymptotic behavior, and explores how end subgraphs influence the ground state degeneracy in the Potts antiferromagnet model.

Contribution

It extends previous calculations of chromatic polynomials to more general strip graphs with different end subgraphs and boundary conditions, and analyzes their asymptotic limits and complex plane properties.

Findings

The asymptotic function W is analytic except on a locus B in the complex plane.

The locus B depends on the end subgraphs J and I, and can enclose regions in the complex plane.

Boundary conditions and end subgraphs significantly affect the ground state degeneracy limits.

Abstract

We calculate the chromatic polynomials $P$ for $n$-vertex strip graphs of the form $J(\prod_{\ell=1}^m H)I$, where $J$ and $I$ are various subgraphs on the left and right ends of the strip, whose bulk is comprised of $m$-fold repetitions of a subgraph $H$. The strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction. This extends our earlier calculations for strip graphs of the form $(\prod_{\ell=1}^m H)I$. We use a generating function method. From these results we compute the asymptotic limiting function $W=\lim_{n \to \infty}P^{1/n}$; for $q \in {\mathbb Z}_+$ this has physical significance as the ground state degeneracy per site (exponent of the ground state entropy) of the $q$-state Potts antiferromagnet on the given strip. In the complex $q$ plane, $W$ is an analytic function except on a certain continuous locus ${\cal B}$. In contrast to the $(\prod_{\ell=1}^m H)I$ strip graphs, where ${\cal B}$ (i) is independent of $I$, and (ii) consists of arcs and possible line segments that do not enclose any regions in the $q$ plane, we find that for some $J(\prod_{\ell=1}^m H)I$ strip graphs, ${\cal B}$ (i) does depend on $I$ and $J$, and (ii) can enclose regions in the $q$ plane. Our study elucidates the effects of different end subgraphs $I$ and $J$ and of boundary conditions on the infinite-length limit of the strip graphs.