Characterization of graphs whose a small power of their edge ideals has a linear free resolution

Let $I(G)$ be the edge ideal of a simple graph $G$. We prove that $I(G{)}^2$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G)\le 3$. Similarly, we show that $I(G{)}^3$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G)\le 4$. We deduce these characterizations from a general formula for the regularity of powers of edge ideals of gap-free graphs $$\mathrm{r}\mathrm{e}\mathrm{g}(I(G{)}^s)=\max (\mathrm{r}\mathrm{e}\mathrm{g}I(G)+s-1,2s),$$ for $s=2,3$.