Depth stability of edge ideals

TL;DR

This paper investigates the depth stability of edge ideals of graphs, establishing bounds on the stabilization point and exploring specific cases like trees with precise parameter relationships.

Contribution

It proves that the depth stability index is always less than the analytic spread and provides a sharper bound for trees, also demonstrating the possible range of these invariants.

Findings

stab(I_G) < \u001ell(I_G) for all graphs

A stronger upper bound for stab(I_G) in trees

Existence of trees with stab and ll values as any two integers with 1 stab 

Abstract

Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $\depth S/I_G^k$ stabilizes is denoted by $\dstab(I_G)$. We show that $\dstab(I_G)<\ell(I_G)$ where $\ell(I_G)$ denotes the analytic spread of $I$. For trees we give a stronger upper bound for $\dstab(I_G)$. We also show for any two integers $1\leq a<b$ there exists a tree for which $\dstab(I_G)=a$ and $\ell(I_G)=b$.