Regularity of powers of edge ideals of unicyclic 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 unicyclic graph, then, for all $s \geq 1$, the regularity of $I(G)^s$ is exactly $2s+\DeclareMathOperator{reg} (I(G))-2$. We also give a combinatorial characterization of unicyclic graphs with regularity $\nu (G)+1$ and $\nu (G)+2$, where $\nu (G)$ denotes the induced matching number of $G$.