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Chromatic roots are dense in the whole complex plane | | Alan D. Sokal
; | | Date: |
19 Dec 2000 | | Journal: | Combin. Probab. Comput. 13, 221-261 (2004) | | Subject: | Statistical Mechanics; Mathematical Physics; Combinatorics; Complex Variables | cond-mat.stat-mech math-ph math.CO math.CV math.MP | | Abstract: | I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs Theta^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. | | Source: | arXiv, cond-mat/0012369 | | Services: | Forum | Review | PDF | Favorites | |
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