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

Abstract

Let $G$ be a finite group. Define the graph $\mathrm{\Delta }\left(G\right)$ on the set $G^{\#}=G\setminus id$ by putting an edge between distinct elements $x,y\in G^{\#}$ if and only if $⟨x,y⟩$ is cyclic. The graph $\mathrm{\Delta }\left(G\right)$ has appeared in the literature under the names cyclic graph and deleted enhanced power graph. We focus on $\mathrm{\Delta }\left(G\times H\right)$ for nontrivial groups $G$ and $H$. In particular, when $\mathrm{\Delta }\left(G\times H\right)$ 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 $\mathrm{\Delta }\left(G\times H\right)$. Finally, we study the connectedness and diameter of $\mathrm{\Delta }\left(G\right)$ in the case that $G$ satisfies certain centralizer conditions.