Depth and Singular Varieties of Exterior Edge Ideals

TL;DR

This paper explores the properties of edge ideals over exterior algebras, establishing bounds on their depth, analyzing specific graph families, and examining the impact of whiskering on depth.

Contribution

It introduces the study of edge ideals in exterior algebras, providing bounds on depth and analyzing effects of graph modifications, which are new contributions in this area.

Findings

Upper bound on depth for general graphs

Refined bound for bipartite graphs, shown to be tight

Depth calculations for cycles, multipartite, spider, and Ferrers graphs

Abstract

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.