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

Abstract

Let $G$ be a finite simple graph and $I\left(G\right)$ denote the corresponding edge ideal. We prove that if $G$ is a very well-covered graph then for all $s\ge 1$ the regularity of $I{\left(G\right)}^s$ is exactly $2s+\nu \left(G\right)-1$, where $\nu \left(G\right)$ denotes the induced matching number of $G$.