Upper bounds for the regularity of powers of edge ideals of graphs

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of $I(G)^q$ in terms of certain combinatorial invariants associated with $G$. We also prove a weaker version of a conjecture by Alilooee, Banerjee, Beyarslan and H\`a on an upper bound for the regularity of $I(G)^q$ and we prove the conjectured upper bound for the class of vertex decomposable graphs. Using these results, we explicitly compute the regularity of $I(G)^q$ for several classes of graphs.