Chromatic roots are dense in the whole complex plane

Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    I show that the zeros of the chromatic polynomials P G(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc |q - 1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z G(q,v) outside the disc |3q + v| < |v|. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

    Original languageEnglish (US)
    Pages (from-to)221-261
    Number of pages41
    JournalCombinatorics Probability and Computing
    Volume13
    Issue number2
    DOIs
    StatePublished - Mar 2004

    Fingerprint

    Argand diagram
    Polynomials
    Roots
    Tutte Polynomial
    Chromatic Polynomial
    Polynomial Model
    Limit Set
    Potts Model
    Zero
    Potts model
    Partition Function
    Planar graph
    Exception
    Analytic function
    Corollary
    Polynomial
    Graph in graph theory
    Theorem

    ASJC Scopus subject areas

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

    Cite this

    Chromatic roots are dense in the whole complex plane. / Sokal, Alan D.

    In: Combinatorics Probability and Computing, Vol. 13, No. 2, 03.2004, p. 221-261.

    Research output: Contribution to journalArticle

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