  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.
