  Edge ideals of finite simple graphs are well-studied over polynomial rings.
In this paper, we initiate the study of edge ideals over exterior algebras,
specifically focusing on the depth and singular varieties of such ideals. We
prove an upper bound on the depth of the edge ideal associated to a general
graph and a more refined bound for bipartite graphs, and we show that both are
tight. We also compute the depth of several large families of graphs including
cycles, complete multipartite graphs, spider graphs, and Ferrers graphs.
Finally, we focus on the effect whiskering a graph has on the depth of the
associated edge ideal.
