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 Hà 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.