A Lower Bound For Depths of Powers of Edge Ideals

TL;DR

This paper establishes lower bounds for the depth of the first three powers of edge ideals of graphs, relating these bounds to the graph's diameter and number of components, advancing understanding of algebraic properties linked to graph structure.

Contribution

It provides explicit lower bounds for the depth of the first three powers of edge ideals based on graph diameter and connectivity, a novel connection between combinatorial and algebraic properties.

Findings

Depth bounds for $I^t$ with $t=1,2,3$ in terms of diameter and components

Explicit formula: $ ext{depth} R/I^t \,\geq\, \lceil\frac{d-4t+5}{3}\rceil + p - 1$

Results extend understanding of algebraic invariants of edge ideals in relation to graph parameters.

Abstract

Let $G$ be a graph and let $I$ be the edge ideal of $G$. Our main results in this article provide lower bounds for the depth of the first three powers of $I$ in terms of the diameter of $G$. More precisely, we show that $\depth R/I^t \geq \left\lceil{\frac{d-4t+5}{3}} \right\rceil +p-1$, where $d$ is the diameter of $G$, $p$ is the number of connected components of $G$ and $1 \leq t \leq 3$. For general powers of edge ideals we show