Chromatic polynomials, Potts models and all that

Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1|<1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.

    Original languageEnglish (US)
    Pages (from-to)324-332
    Number of pages9
    JournalPhysica A: Statistical Mechanics and its Applications
    Volume279
    Issue number1
    DOIs
    StatePublished - May 1 2000

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    Chromatic Polynomial
    Potts Model
    Partition Function
    polynomials
    partitions
    Zero
    Polynomial
    Finite Graph
    Graph in graph theory
    Maximum Degree
    Planar graph
    Countable
    Upper bound
    Arbitrary

    ASJC Scopus subject areas

    • Mathematical Physics
    • Statistical and Nonlinear Physics

    Cite this

    Chromatic polynomials, Potts models and all that. / Sokal, Alan D.

    In: Physica A: Statistical Mechanics and its Applications, Vol. 279, No. 1, 01.05.2000, p. 324-332.

    Research output: Contribution to journalArticle

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