### 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) |
---|---|

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 2009 |

### Fingerprint

### 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

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

### Cite this

*Journal of Combinatorial Theory, Series B*,

*99*(6), 869-903. https://doi.org/10.1016/j.jctb.2009.03.002

**Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids.** / Jackson, Bill; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series B*, vol. 99, no. 6, pp. 869-903. https://doi.org/10.1016/j.jctb.2009.03.002

}

TY - JOUR

T1 - Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

AU - Jackson, Bill

AU - Sokal, Alan D.

PY - 2009/11

Y1 - 2009/11

N2 - 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.

AB - 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.

KW - Characteristic polynomial

KW - Chromatic polynomial

KW - Chromatic root

KW - Dichromatic polynomial

KW - Flow polynomial

KW - Flow root

KW - Graph

KW - Matroid

KW - Potts model

KW - Tutte polynomial

KW - Zero-free interval

UR - http://www.scopus.com/inward/record.url?scp=72049120149&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72049120149&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2009.03.002

DO - 10.1016/j.jctb.2009.03.002

M3 - Article

VL - 99

SP - 869

EP - 903

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 6

ER -