Regularity of powers of edge ideals

Let $G$ be a graph and $I=I(G)$ be its edge ideal. In this talk, we review the main results about the regularity of powers of $I$. Recently, Beyarslam, H${\rm \grave{a}}$ and Trung proved that ${\rm reg}(I(G)^{s})\geq 2s+{\rm indmatch}(G)-1$, for every $s\geq 1$. They also proved that the equality happens for every $s\geq 1$, when $G$ is a forest and for every $s\geq 2$, when $G$ is a cycle. In this talk, we determine a new class of graphs for which the equality holds. More explicit, we show that if $G$ is a whiskered cycle graph, then ${\rm reg}(I^{s})=2s+\lceil \frac{n-1}{2}\rceil-1$, for every integer $s\geq 1$.