Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits

TL;DR

This paper computes chromatic polynomials for strip graphs, analyzes their asymptotic limits, and explores the nonanalytic locus in the complex plane, providing insights into lattice properties and connections to the Potts model.

Contribution

It introduces a generating function method to determine chromatic polynomials and their asymptotic limits for strip graphs, linking combinatorics with statistical mechanics.

Findings

Exact locus of nonanalytic points ${\

} in the complex plane identified.

As strip width increases, the loci elongate and approach configurations resembling 2D lattices.

Abstract

We calculate the chromatic polynomials $P((G_s)_m,q)$ and, from these, the asymptotic limiting functions $W(\{G_s\},q)=\lim_{n \to \infty}P(G_s,q)^{1/n}$ for families of $n$-vertex graphs $(G_s)_m$ comprised of $m$ repeated subgraphs $H$ adjoined to an initial graph $I$. These calculations of $W(\{G_s\},q)$ for infinitely long strips of varying widths yield important insights into properties of $W(\Lambda,q)$ for two-dimensional lattices $\Lambda$. In turn, these results connect with statistical mechanics, since $W(\Lambda,q)$ is the ground state degeneracy of the $q$-state Potts model on the lattice $\Lambda$. For our calculations, we develop and use a generating function method, which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant $W(\{G_s\},q)$ function in the limit $n \to \infty$. From this, we obtain the exact continuous locus of points ${\cal B}$ where $W(\{G_s\},q)$ is nonanalytic in the complex $q$ plane. This locus is shown to consist of arcs which do not separate the $q$ plane into disconnected regions. Zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities ${\cal B}$. We find that as the width of the infinitely long strips is increased, the arcs comprising ${\cal B}$ elongate and move toward each other, which enables one to understand the origin of closed regions that result for the (infinite) 2D lattice.