Summary: Let \(G\) be a finite group. Define the graph \(\Delta (G)\) on the set \(G^{\#}=G\backslash\) id by putting an edge between distinct elements \(x,y \in G^{\#}\) if and only if \(\langle x,y\rangle\) is cyclic. The graph \(\Delta(G)\) has appeared in the literature under the names cyclic graph and deleted enhanced power graph. We focus on \(\Delta(G\times H)\) for nontrivial groups \(G\) and \(H\). In particular, when \(\Delta(G\times H)\) is connected, we obtain an upper bound on the diameter; in addition, we present an example meeting this bound. Also, we establish necessary and sufficient conditions for the connectedness of \(\Delta(G\times H)\). Finally, we study the connectedness and diameter of \(\Delta (G)\) in the case that \(G\) satisfies certain centralizer conditions.