Families of graphs with [Formula Presented] functions that are nonanalytic at [Formula Presented]

Denoting [Formula Presented] as the chromatic polynomial for coloring an [Formula Presented]-vertex graph [Formula Presented] with [Formula Presented] colors, and considering the limiting function [Formula Presented], a fundamental question in graph theory is the following: is [Formula Presented] analytic or not at the origin of the [Formula Presented] plane (where the complex generalization of [Formula Presented] is assumed)? This question is also relevant in statistical mechanics because [Formula Presented], where [Formula Presented] is the ground state entropy of the [Formula Presented]-state Potts antiferromagnet on the lattice graph [Formula Presented], and the analyticity of [Formula Presented] at [Formula Presented] is necessary for the large-[Formula Presented] series expansions of [Formula Presented]. Although [Formula Presented] is analytic at [Formula Presented] for many [Formula Presented], there are some [Formula Presented] for which it is not; for these, [Formula Presented] has no large-[Formula Presented] series expansion. It is important to understand the reason for this nonanalyticity. Here we give a general condition that determines whether or not a particular [Formula Presented] is analytic at [Formula Presented] and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with [Formula Presented] functions that are nonanalytic at [Formula Presented] and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for [Formula Presented] to be analytic at [Formula Presented] is that [Formula Presented] is a regular lattice graph [Formula Presented]. (This is known not to be a necessary condition.)