The cyclic graph (deleted enhanced power graph) of a direct product

Let $G$ be a finite group. Define graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to adjacent if and only $\langle x,y\rangle$ is cyclic. Denote this $\Delta(G)$. The $\Delta(G)$ has appeared in literature under names cyclic deleted enhanced power graph. If $H$ are nontrivial groups, then $\Delta(G\times H)$ completely characterized. In particular, connected, diameter bound obtained, along with an example meeting bound. Also, necessary sufficient conditions for disconnectedness of established.