Exact chromatic polynomials for toroidal chains of complete graphs

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Abstract

We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs Kb with b=5, 6 which have periodic or twisted periodic boundary condition in the longitudinal direction. In the Lx → ∞ 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 b-1 and the explicit forms of some classes of eigenvalues. We prove that the maximal point where ℬ crosses the real axis, qc, satisfies the inequality qc ≤ 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.

Original languageEnglish
Pages (from-to)397-426
Number of pages30
JournalPhysica A: Statistical Mechanics and its Applications
Volume313
Issue number3-4
DOIs
Publication statusPublished - 2002 Oct 15

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Condensed Matter Physics

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