Dimension, depth and zero-divisors of the algebra of basic $k$-covers of a graph

We study the basic $k$-covers of a bipartite graph $G$; the algebra $\text{AG}$ they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when $\text{AG}$ is a domain in terms of the combinatorics of $G$; if follows from a result of Hochster that when $\text{AG}$ is a domain, it is also Cohen-Macaulay. We then study the dimension of $\text{AG}$ by introducing a geometric invariant of bipartite graphs, the “graphical dimension”. We show that the graphical dimension of $G$ is not larger than $\dim (\text{AG})$, 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.