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

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

doi:10.2140/involve.2021.14.167

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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

DOI: 10.2140/involve.2021.14.167

Abstract

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

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Keywords

cyclic graphs, deleted enhanced power graphs, direct products

Mathematical Subject Classification

Primary: 20D40

Secondary: 05C25

Milestones

Received: 3 October 2020

Revised: 30 November 2020

Accepted: 4 December 2020

Published: 4 March 2021

Communicated by Scott T. Chapman

Authors

David G. Costanzo
Department of Mathematical Sciences Kent State University Kent, OH United States

Mark L. Lewis
Department of Mathematical Sciences Kent State University Kent, OH United States

Stefano Schmidt
Department of Mathematics Columbia University New York, NY United States

Eyob Tsegaye
Department of Mathematics Stanford University Stanford, CA United States

Gabe Udell
Department of Mathematics Pomona College Claremont, CA United States

Vol. 14, No. 1, 2021