Sokal, AD;
(2004)
Chromatic roots are dense in the whole complex plane.
COMB PROBAB COMPUT
, 13
(2)
221 - 261.
10.1017/S0963548303006023.
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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 - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots 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 oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.
| Type: | Article |
|---|---|
| Title: | Chromatic roots are dense in the whole complex plane |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1017/S0963548303006023 |
| Keywords: | GROUND-STATE ENTROPY, MODEL PARTITION-FUNCTIONS, ANTIFERROMAGNETIC POTTS MODELS, PERIODIC BOUNDARY-CONDITIONS, HYPERBOLIC COXETER GROUPS, NONCOMPACT W BOUNDARIES, RANDOM-CLUSTER MEASURES, CYCLIC STRIP GRAPHS, SQUARE-LATTICE, TRIANGULAR-LATTICE |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/9064 |
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