Ground-state degeneracy of Potts antiferromagnets: Homeomorphic classes with noncompact W boundaries

We present exact calculations of the zero-temperature partition function Z(G, q, T = O) and ground-state degeneracy W({G},q) for the q-stats Potts antiferromagnet on a number of families of graphs G for which (generalizing q from ℤ₊ to ℂ) the boundary ℬ of regions of analyticity of W in the complex q plane is noncompact, passing through z = 1/q = 0. For these types of graphs, since the reduced function W_red = q^-1W is nonanalytic at z = 0, there is no large-q Taylor series expansion of W_red The study of these graphs thus gives insight into the conditions for the validity of the large-q expansions. It is shown how such (families of) graphs can be generated from known families by homeomorphic expansion.