Bounds for the regularity of product of edge ideals

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 $J_1,\dots ,J_d$ 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.