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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
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 ggk 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.

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