### Abstract

We find zero-free regions in the complex plane at large |. q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w={we}. This generalizes a result of Sokal (2001) [28] that applies only within the complex antiferromagnetic regime |1+we|≤1. Our proof uses the polymer-gas representation of the multivariate Tutte polynomial together with the Penrose identity.

Original language | English (US) |
---|---|

Pages (from-to) | 21-45 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 103 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2013 |

### Fingerprint

### Keywords

- Chromatic polynomial
- Graph
- Lambert w function
- Multivariate tutte polynomial
- Penrose identity
- Penrose inequality
- Potts model

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Theory, Series B*,

*103*(1), 21-45. https://doi.org/10.1016/j.jctb.2012.08.002

**Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights.** / Jackson, Bill; Procacci, Aldo; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series B*, vol. 103, no. 1, pp. 21-45. https://doi.org/10.1016/j.jctb.2012.08.002

}

TY - JOUR

T1 - Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights

AU - Jackson, Bill

AU - Procacci, Aldo

AU - Sokal, Alan D.

PY - 2013/1

Y1 - 2013/1

N2 - We find zero-free regions in the complex plane at large |. q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w={we}. This generalizes a result of Sokal (2001) [28] that applies only within the complex antiferromagnetic regime |1+we|≤1. Our proof uses the polymer-gas representation of the multivariate Tutte polynomial together with the Penrose identity.

AB - We find zero-free regions in the complex plane at large |. q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w={we}. This generalizes a result of Sokal (2001) [28] that applies only within the complex antiferromagnetic regime |1+we|≤1. Our proof uses the polymer-gas representation of the multivariate Tutte polynomial together with the Penrose identity.

KW - Chromatic polynomial

KW - Graph

KW - Lambert w function

KW - Multivariate tutte polynomial

KW - Penrose identity

KW - Penrose inequality

KW - Potts model

UR - http://www.scopus.com/inward/record.url?scp=84869888994&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869888994&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2012.08.002

DO - 10.1016/j.jctb.2012.08.002

M3 - Article

AN - SCOPUS:84869888994

VL - 103

SP - 21

EP - 45

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -