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