On Oriented Diameter of Power Graphs

In this paper, we study the oriented diameter of power graphs of groups. We show that a \(2\)-edge connected power graph of a finite group has oriented diameter at most \(4\). We prove that the power graph of the cyclic group of order \(n\) has oriented diameter \(2\) for all \(n\neq 1,2,4,6\). For...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Authors: Benson, Deepu, Das, Bireswar, Dey, Dipan, Ghosh, Jinia
Format: Article
Language:English
Subjects:
Algorithms
Completeness
Graphs
Group theory
Polynomials
Online Access:Get full text
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Summary:In this paper, we study the oriented diameter of power graphs of groups. We show that a \(2\)-edge connected power graph of a finite group has oriented diameter at most \(4\). We prove that the power graph of the cyclic group of order \(n\) has oriented diameter \(2\) for all \(n\neq 1,2,4,6\). For non-cyclic finite nilpotent groups, we show that the oriented diameter of corresponding power graphs is at least \(3\). Moreover, we provide necessary and sufficient conditions for the oriented diameter of \(2\)-edge connected power graphs of finite non-cyclic nilpotent groups to be either \(3\) or \(4\). This, in turn, gives an algorithm for computing the oriented diameter of the power graph of a given nilpotent group that runs in time polynomial in the size of the group.
ISSN:2331-8422