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

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 $Δ(G)$. The graph $Δ(G)$ has appeared in the literature under the names cyclic graph and deleted enhanced power graph. If $G$ and $H$ are nontrivial groups, then $Δ(G\times H)$ is completely characterized. In particular, if $Δ(G\times H)$ is connected, then a diameter bound is obtained, along with an example meeting this bound. Also, necessary and sufficient conditions for the disconnectedness of $Δ(G\times H)$ are established.