  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.
