Depths of Powers of the Edge Ideal of a Tree

TL;DR

This paper establishes lower bounds for the depths of powers of the edge ideal of a tree or forest, relating these bounds to the graph's diameter and number of components, thus informing when depth stabilization occurs.

Contribution

It provides new lower bounds for the depths of powers of edge ideals of trees and forests based on their diameters and components.

Findings

Lower bounds for depths of R/I^t in terms of tree diameter

Bounds applicable to forests with multiple components

Insights into depth stabilization for powers of edge ideals

Abstract

Lower bounds are given for the depths of R/I^t for t at least one when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.