### 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) |
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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 |

### Keywords

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

### ASJC Scopus subject areas

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

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## Cite this

Jackson, B., Procacci, A., & Sokal, A. D. (2013). Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights.

*Journal of Combinatorial Theory. Series B*,*103*(1), 21-45. https://doi.org/10.1016/j.jctb.2012.08.002