Exact chromatic polynomials for toroidal chains of complete graphs
Abstract
We present exact calculations of the partition function of the zerotemperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs K_{b} with b=5,6 which have periodic or twisted periodic boundary condition in the longitudinal direction. In the L_{x}→∞ limit, the continuous accumulation set of the chromatic zeros B is determined. We give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b1 and the explicit forms of some classes of eigenvalues. We prove that the maximal point where B crosses the real axis, q_{c}, satisfies the inequality q_{c}⩽ b for 2⩽ b, the minimum value of q at which B crosses the real q axis is q=0, and we make a conjecture concerning the structure of the chromatic polynomial for Klein bottle strips.
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 October 2002
 DOI:
 10.1016/S03784371(02)009779
 arXiv:
 arXiv:mathph/0111028
 Bibcode:
 2002PhyA..313..397C
 Keywords:

 Mathematical Physics
 EPrint:
 36 pages, latex, 2 postscript figures included