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Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

Published online by Cambridge University Press:  01 March 2008

ROBERTO FERNÁNDEZ  and
ALDO PROCACCI
ROBERTO FERNÁNDEZ
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP.12, 76801 Saint Etienne du Rouvray, France (e-mail: [email protected])
ALDO PROCACCI
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP.12, 76801 Saint Etienne du Rouvray, France (e-mail: [email protected]) Departamento de Matemática-ICEx, UFMG, CP 702, Belo Horizonte MG 30.161-970, Brazil (e-mail: [email protected])
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Abstract

We prove that the chromatic polynomial of a finite graph of maximal degree Δ is free of zeros for |q| ≥ C*(Δ) with This improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 2 , March 2008 , pp. 225 - 238
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Borgs, C. (2006) Absence of zeros for the chromatic polynomial on bounded degree graphs. Combin. Probab. Comput. 15 63–74.CrossRefGoogle Scholar
[2]Brydges, D. C. (1984) A short cluster in cluster expansions. In Critical Phenomena, Random Systems, Gauge Theories (Osterwalder, K. and Stora, R., eds), Elsevier, pp. 129–83.Google Scholar
[3]Cammarota, C. (1982) Decay of correlations for infinite range interactions in unbounded spin systems. Comm. Math. Phys. 85 517–28.CrossRefGoogle Scholar
[4]Dobrushin, R. L. (1996) Estimates of semi-invariants for the Ising model at low temperatures. In Topics in Statistics and Theoretical Physics. Amer. Math. Soc. Transl. Ser. 2 177 59–81.Google Scholar
[5]Gruber, C. and Kunz, H. (1971) General properties of polymer systems. Comm. Math. Phys. 22 133–61.CrossRefGoogle Scholar
[6]Fernández, R. and Procacci, A. (2007) Cluster expansions for abstract polymer models: New bounds from an old approach. Preprint arXiv math-ph/0605041; Comm. Math. Phys. 274 123–40.Google Scholar
[7]Kotecký, R. and Preiss, D. (1986) Cluster expansion for abstract polymer models. Comm. Math. Phys. 103 491–498.CrossRefGoogle Scholar
[8]Penrose, O. (1967) Convergence of fugacity expansions for classical systems. In Statistical Mechanics: Foundations and Applications (Bak, A., ed.), Benjamin, New York.Google Scholar
[9]Procacci, A. and Scoppola, B. (1999) Polymer gas approach to N-body lattice systems. J. Statist. Phys. 96 49–68.CrossRefGoogle Scholar
[10]Procacci, A., Scoppola, B. and Gerasimov, V. (2003) Potts model on infinite graphs and the limit of chromatic polynomials. Comm. Math. Phys. 235 215–31.CrossRefGoogle Scholar
[11]Ruelle, D. (1969) Statistical Mechanics: Rigorous Results, Benjamin, New York/Amsterdam.Google Scholar
[12]Sokal, A. (2001) Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combin. Probab. Comput. 10 41–77.CrossRefGoogle Scholar