Depth of powers of edge ideals of cycles and trees

TL;DR

This paper determines the depth of powers of edge ideals for cycles and certain trees, providing explicit formulas that advance understanding of algebraic properties related to graph structures.

Contribution

It introduces explicit formulas for the depth of powers of edge ideals of cycles and starlike trees, extending algebraic graph theory knowledge.

Findings

Depth formula for cycle edge ideals for 2 ≤ t < (n+1)/2

Depth formula for powers of starlike tree edge ideals

Provides algebraic insights into graph structures

Abstract

Let $I$ be the edge ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. We prove that for $2 \le t < \lceil (n+1)/2 \rceil$, $$\operatorname{depth} (S/I^t) = \lceil \frac{n -t + 1}{3} \rceil.$$ When $G = T_{\mathbf{a}}$ is a starlike tree which is the join of $k$ paths of length $a_1, \ldots, a_k$ at a common root $1$, we give a formula for the depth of powers of $I(T_{\mathbf{a}})$.