Chromatic polynomials of random graphs

TL;DR

This paper investigates the distribution of chromatic polynomial zeros in random graphs up to 30 vertices, revealing empirical scaling laws that connect these zeros to thermodynamic properties.

Contribution

It presents the first large-scale analysis of chromatic zeros in random graphs across all edge densities, utilizing recent algorithms for polynomial computation.

Findings

Chromatic zeros exhibit a consistent layout in the complex plane.

The crossing point scales linearly with average degree.

Implications for phase transitions and thermodynamic entropy in graph systems.

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.