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

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

1999-04-11

arxiv.org/abs/cond-mat/9904146v3

Comb.Probab.Comput. 10 (2001) 41-77

Physics

Condensed Matter

Statistical Mechanics

47 pages (LaTeX). Revised version contains slightly simplified proofs of Propositions 4.2 and 4.5. Version 3 fixes a silly err

Scientific paper

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. 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 $|1 + v_e| \le 1$. 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-Koteck\'y-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs $G$ of second-largest degree $\le r$, 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-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.

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