  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.
