#### Vol. 14, No. 1, 2021

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

### David G. Costanzo, Mark L. Lewis, Stefano Schmidt, Eyob Tsegaye and Gabe Udell

Vol. 14 (2021), No. 1, 167–179
##### Abstract

Let $G$ be a finite group. Define the graph $\Delta \left(G\right)$ 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 $\Delta \left(G\right)$ has appeared in the literature under the names cyclic graph and deleted enhanced power graph. We focus on $\Delta \left(G×H\right)$ for nontrivial groups $G$ and $H$. In particular, when $\Delta \left(G×H\right)$ 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 $\Delta \left(G×H\right)$. Finally, we study the connectedness and diameter of $\Delta \left(G\right)$ in the case that $G$ satisfies certain centralizer conditions.

##### Keywords
cyclic graphs, deleted enhanced power graphs, direct products
Primary: 20D40
Secondary: 05C25