### Abstract

The chromatic polynomial P_{G} (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 Z_{G} (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 language | English (US) |
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Pages (from-to) | 869-903 |

Number of pages | 35 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 99 |

Issue number | 6 |

DOIs | |

State | Published - 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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## Cite this

Jackson, B., & Sokal, A. D. (2009). Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids.

*Journal of Combinatorial Theory. Series B*,*99*(6), 869-903. https://doi.org/10.1016/j.jctb.2009.03.002