Linear polynomial for the regularity of powers of edge ideals of very well-covered graphs

TL;DR

This paper establishes a precise formula for the regularity of powers of edge ideals in very well-covered graphs, showing a linear relationship with the power degree and the graph's matching number.

Contribution

It provides an exact regularity formula for all powers of edge ideals in very well-covered graphs, a class not previously characterized in this way.

Findings

Regularity of $I(G)^s$ is exactly $2s + u(G) - 1$ for all $s \, \geq \, 2$.

The result confirms a linear pattern in the regularity growth of powers of edge ideals.

The work extends understanding of algebraic invariants in graph theory, specifically for very well-covered graphs.

Abstract

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper we prove that if $G$ is a very well-covered graph then for all $s \geq 2$ the regularity of $I(G)^s$ is exactly $2s+\nu(G)-1$.