Regularity of Powers of Unicyclic Graphs

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 $s \geq 1$theregularityof$I(G)^s$isexactly$2s+$\text{reg}$(I(G))-2$. We also characterize the unicyclic graphs with regularity $\nu (G)+1$ and $\nu (G)+2$, where $\nu (G)$ denotes the induced matching number of $G$.