Published online by Cambridge University Press: 07 June 2013
For a countable discrete space $V$, every nondegenerate separable
${C}^{\ast } $-correspondence over
${c}_{0} (V)$ is isomorphic to one coming from a directed graph with vertex set
$V$. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of
${C}^{\ast } $-correspondences) and for topological graphs (where
$V$ is locally compact Hausdorff), and we discuss the obstructions that arise.