On Oriented Diameter of Power Graphs

TL;DR

This paper investigates the oriented diameter of power graphs of finite groups, establishing bounds and exact values for various classes, and provides an efficient algorithm for computing this diameter in nilpotent groups.

Contribution

It offers new bounds and exact values for the oriented diameter of power graphs of finite groups, especially cyclic and nilpotent groups, and introduces a polynomial-time algorithm for computation.

Findings

Power graph of a 2-edge connected finite group has oriented diameter ≤ 4.

Cyclic groups of order n have oriented diameter 2 for all n ≠ 1,2,4,6.

For non-cyclic finite nilpotent groups, the oriented diameter is at least 3.

Abstract

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.