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

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