### Abstract

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1|<1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.

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

Pages (from-to) | 324-332 |

Number of pages | 9 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 279 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2000 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*279*(1), 324-332. https://doi.org/10.1016/S0378-4371(99)00519-1

**Chromatic polynomials, Potts models and all that.** / Sokal, Alan D.

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 279, no. 1, pp. 324-332. https://doi.org/10.1016/S0378-4371(99)00519-1

}

TY - JOUR

T1 - Chromatic polynomials, Potts models and all that

AU - Sokal, Alan D.

PY - 2000/5/1

Y1 - 2000/5/1

N2 - The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1|<1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.

AB - The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1|<1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.

UR - http://www.scopus.com/inward/record.url?scp=0033727834&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033727834&partnerID=8YFLogxK

U2 - 10.1016/S0378-4371(99)00519-1

DO - 10.1016/S0378-4371(99)00519-1

M3 - Article

VL - 279

SP - 324

EP - 332

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 1

ER -