  Let $I$ and $J$ be edge ideals in a polynomial ring $R =
\mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$. In this paper, we obtain a
general upper and lower bound for the Castelnuovo-Mumford regularity of $IJ$ in
terms of certain invariants associated with $I$ and $J$. Using these results,
we explicitly compute the regularity of $IJ$ for several classes of edge
ideals. Let $J_1,\ldots,J_d$ be edge ideals in a polynomial ring $R$ with $J_1
\subseteq \cdots \subseteq J_d$. Finally, we compute the precise expression for
the regularity of $J_1 J_2\cdots J_d$ when $d \in \{3,4\}$ and $J_d$ is the
edge ideal of complete graph.
