  We study the basic $k$-covers of a bipartite graph $G$; the algebra $\AG$
they span, first studied by Herzog, is the fiber cone of the Alexander dual of
the edge ideal. We characterize when $\AG$ is a domain in terms of the
combinatorics of $G$; if follows from a result of Hochster that when $\AG$ is a
domain, it is also Cohen-Macaulay. We then study the dimension of $\AG$ by
introducing a geometric invariant of bipartite graphs, the "graphical
dimension". We show that the graphical dimension of $G$ is not larger than
$\dim(\AG)$, and equality holds in many cases (e.g. when $G$ is a tree, or a
cycle). Finally, we discuss applications of this theory to the arithmetical
rank.
