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 Δ(G) on the set G# = G id by putting an edge between distinct elements x,y 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 × H) for nontrivial groups G and H. In particular, when Δ(G × 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 × H). Finally, we study the connectedness and diameter of Δ(G) in the case that G satisfies certain centralizer conditions.

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