Chromatic Polynomials, Potts Models and All That

@misc{citeulike:1645787, abstract = {The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| \< 1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.}, author = {Sokal, Alan D. }, citeulike-article-id = {1645787}, comment = {proof that number of colorings of a graph with k colors is a restriction of a polynomial (chromatic polynomial), real zeros correspond to unsatisfiable graph, shows that no zeros for loopless graphs, bounds the region where zeros must lie in complex plane, why care about complex plane?}, eprint = {cond-mat/9910503}, keywords = {ising}, month = {Oct}, posted-at = {2007-09-11 23:28:35}, priority = {2}, title = {Chromatic Polynomials, Potts Models and All That}, url = {http://arxiv.org/abs/cond-mat/9910503}, year = {1999} }

TY - ELEC ID - citeulike:1645787 L3 - citeulike-article-id:1645787 TI - Chromatic Polynomials, Potts Models and All That N2 - The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| < 1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree. KW - ising N1 - proof that number of colorings of a graph with k colors is a restriction of a polynomial (chromatic polynomial), real zeros correspond to unsatisfiable graph, shows that no zeros for loopless graphs, bounds the region where zeros must lie in complex plane, why care about complex plane? AU - Sokal, Alan PY - 1999/Oct/30/ UR - http://arxiv.org/abs/cond-mat/9910503 ER -