  We present exact calculations of the zero-temperature partition function
$Z(G,q,T=0)$ and ground-state degeneracy $W(\{G\},q)$ for the $q$-state Potts
antiferromagnet on a number of families of graphs $G$ for which (generalizing
$q$ from ${\mathbb Z}_+$ to ${\mathbb C}$) the boundary ${\cal B}$ 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
$W_{red.}=q^{-1}W$ 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.
