### Abstract

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial P
_{G}(q) lie in the disc |q| < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function Z
_{G}(q, {v
_{e}}) in the complex antiferromagnetic regime |1 + v
_{e}| ≤ 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of Z
_{G}(q, {v
_{e}}) to a polymer gas, followed by verification of the Dobrushin-Kotecký-Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree ≤ r, the zeros of P
_{G}(q) lie in the disc |q| < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colboum conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

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

Pages (from-to) | 41-77 |

Number of pages | 37 |

Journal | Combinatorics Probability and Computing |

Volume | 10 |

Issue number | 1 |

State | Published - 2001 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Statistics and Probability
- Theoretical Computer Science

### Cite this

*Combinatorics Probability and Computing*,

*10*(1), 41-77.

**Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions.** / Sokal, Alan D.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 10, no. 1, pp. 41-77.

}

TY - JOUR

T1 - Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

AU - Sokal, Alan D.

PY - 2001

Y1 - 2001

N2 - We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial P G(q) lie in the disc |q| < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function Z G(q, {v e}) in the complex antiferromagnetic regime |1 + v e| ≤ 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of Z G(q, {v e}) to a polymer gas, followed by verification of the Dobrushin-Kotecký-Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree ≤ r, the zeros of P G(q) lie in the disc |q| < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colboum conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

AB - We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial P G(q) lie in the disc |q| < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function Z G(q, {v e}) in the complex antiferromagnetic regime |1 + v e| ≤ 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of Z G(q, {v e}) to a polymer gas, followed by verification of the Dobrushin-Kotecký-Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree ≤ r, the zeros of P G(q) lie in the disc |q| < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colboum conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

UR - http://www.scopus.com/inward/record.url?scp=0035608319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035608319&partnerID=8YFLogxK

M3 - Article

VL - 10

SP - 41

EP - 77

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1

ER -