Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

Bill Jackson, Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    The chromatic polynomial PG (q) of a loopless graph G is known to be non-zero (with explicitly known sign) on the intervals (- ∞, 0), (0, 1) and (1, 32 / 27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial ZG (q, v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.

    Original languageEnglish (US)
    Pages (from-to)869-903
    Number of pages35
    JournalJournal of Combinatorial Theory. Series B
    Volume99
    Issue number6
    DOIs
    StatePublished - Nov 1 2009

    Keywords

    • Characteristic polynomial
    • Chromatic polynomial
    • Chromatic root
    • Dichromatic polynomial
    • Flow polynomial
    • Flow root
    • Graph
    • Matroid
    • Potts model
    • Tutte polynomial
    • Zero-free interval

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Discrete Mathematics and Combinatorics
    • Computational Theory and Mathematics

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