  Let $I(G)$ be the edge ideal of a bicyclic graph. In this paper, we
characterize the Castelnuovo-Mumford regularity of $I(G)$ in terms of the
induced matching number of $G$. For the base case of this family of graphs,
i.e. dumbbell graph, we explicitly compute the induced matching number.
Moreover, we prove that $
\operatorname{reg}(I(G)^q)=2q+\operatorname{reg}(I(G))-2$, for all $ q\geq 1 $,
when $ G $ is a dumbbell graph with a connecting path having no more than two
vertices.
