  Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is
a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0.
In particular, for these classes of graphs, we provide the asymptotic linear
function reg(I^s) as s > 0, and the initial value of s starting from which
reg(I^s) attains its linear form. We also give new bounds on the regularity of
I when G contains a Hamiltonian path and when G is a Hamiltonian graph.
