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 $⟨x,y⟩$ 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.

Fixing typo in title