Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T18:52:18.236Z Has data issue: false hasContentIssue false

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

Published online by Cambridge University Press:  12 April 2001

ALAN D. SOKAL
Affiliation:
Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA (e-mail: sokal@nyu.edu)

Abstract

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree [les ] r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc [mid ]q[mid ] < C(r). Furthermore, C(r) [les ] 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q, {ve}) in the complex antiferromagnetic regime [mid ]1 + ve[mid ] [les ] 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Kotecký–Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree [les ] r, the zeros of PG(q) lie in the disc [mid ]q[mid ] < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown–Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

Type
Research Article
Copyright
2001 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)