Chromatic roots are dense in the whole complex plane
Sokal, Alan D.
arXiv, 0012369 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2000-12-19
Classification:  Condensed Matter - Statistical Mechanics,  Mathematical Physics,  Mathematics - Combinatorics,  Mathematics - Complex Variables 
@article{0012369,
     author = {Sokal, Alan D.},
     title = {Chromatic roots are dense in the whole complex plane},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0012369}
}

Sokal, Alan D. Chromatic roots are dense in the whole complex plane. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0012369/