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

Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial P G(q) lie in the disc |q| < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function Z G(q, {v e}) in the complex antiferromagnetic regime |1 + v e| ≤ 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of Z G(q, {v e}) 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 ≤ r, the zeros of P G(q) lie in the disc |q| < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colboum conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

    Original languageEnglish (US)
    Pages (from-to)41-77
    Number of pages37
    JournalCombinatorics Probability and Computing
    Volume10
    Issue number1
    StatePublished - 2001

    Fingerprint

    Potts model
    Chromatic Polynomial
    Polynomial Model
    Potts Model
    Partition Function
    Polynomials
    Zero
    Polymers
    Series-parallel Graph
    Graph in graph theory
    Gases
    Maximum Degree
    Corollary
    Polynomial
    Statistical Models

    ASJC Scopus subject areas

    • Computational Theory and Mathematics
    • Mathematics(all)
    • Discrete Mathematics and Combinatorics
    • Statistics and Probability
    • Theoretical Computer Science

    Cite this

    Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions. / Sokal, Alan D.

    In: Combinatorics Probability and Computing, Vol. 10, No. 1, 2001, p. 41-77.

    Research output: Contribution to journalArticle

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