Structural properties of group power digraphs

Grant, Ben; Ireland, Robert; Koenig, E.; Wood, Adam

doi:10.2140/involve.2023.16.809

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Structural properties of group power digraphs

Ben Grant, Robert Ireland, E. Koenig and Adam Wood

Vol. 16 (2023), No. 5, 809–832

DOI: 10.2140/involve.2023.16.809

Abstract

We study group power digraphs $Γ (G, k)$ , where $G$ is a finite group and $k$ is a positive integer. A group power digraph is a directed graph derived from the power function $g \mapsto g^{k}$ on a finite group $G$ . We present results on various properties of these power digraphs, including indegree of vertices, structure of components, existence of cycles, and a certain type of symmetry. Arbitrary finite groups are discussed briefly, but the focus of this paper is on three families of finite groups: cyclic, dihedral, and symmetric groups. In each case, properties specific to the family of groups are presented.

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Keywords

directed graph, finite group, power digraph, $m$-symmetry

Mathematical Subject Classification

Primary: 05C20, 05C25

Milestones

Received: 26 May 2022

Revised: 8 December 2022

Accepted: 9 December 2022

Published: 9 December 2023

Communicated by Ronald Gould

Authors

Ben Grant
Mathematics, Statistics, and Computer Science Department St. Olaf College Northfield, MN United States

Robert Ireland
Mathematics, Statistics, and Computer Science Department St. Olaf College Northfield, MN United States

E. Koenig
Mathematics, Statistics, and Computer Science Department St. Olaf College Northfield, MN United States

Adam Wood
Mathematics, Statistics and Computer Science Department St. Olaf College Northfield, MN United States

Vol. 16, No. 5, 2023