Let $\tilde {C_{n}}$ be the graph by adding an ear to $C_n$ and $I=I(\tilde {C_{n}})$ be its edge ideal. In this paper, we prove that $\operatorname {reg}(I^{s})=2s+\lfloor \frac {n+1}{3}\rfloor -1$ for all $s \geq 1$. Let $G$ be the bicyclic graph $C_m \sqcup C_n$ with edge ideal $I = I(G)$; we compute the regularity of $I^s$ for all $s \geq 1$. In particular, in some cases, we get $\operatorname {reg}(I^{s})=2s+\lfloor \frac {m}{3}\rfloor +\lfloor \frac {n}{3}\rfloor -1$ for all $s \geq 2$.