Chromatic polynomials of random graphs
arXiv:1709.06209 · doi:10.1088/1751-8113/43/17/175002
Abstract
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
arXiv comment13 pages, 5 figures
References in corpus (7)
- Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial
- Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets
- Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
- Chromatic Polynomials, Potts Models and All That
- Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
- Reliability Polynomials and their Asymptotic Limits for Families of Graphs
- Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions