# Powers of Edge Ideals with Linear Resolutions

## TL;DR

This paper proves that for certain classes of graphs, specifically gap-free and diamond-free graphs, all powers of their edge ideals have linear minimal free resolutions, extending understanding of algebraic properties of these ideals.

## Contribution

It establishes that the powers of edge ideals of gap-free and diamond-free graphs have linear resolutions for all powers s ≥ 2, a new result in algebraic combinatorics.

## Key findings

- All powers of edge ideals of gap-free, diamond-free graphs have linear resolutions.
- The result applies for all powers s ≥ 2.
- Provides new insights into the algebraic structure of these graph classes.

## Abstract

We show that if $G$ is a gap-free and diamond-free graph, then $I(G)^s$ has a linear minimal free resolution for every $s\geq 2$.