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On Oriented Diameter of Power Graphs

Deepu Benson, Bireswar Das, Dipan Dey, Jinia Ghosh

TL;DR

This work determines the oriented diameter of power graphs for broad classes of finite groups. It proves a sharp upper bound: any 2-edge-connected power graph Pow(G) has OD ≤ 4, and it provides precise OD values for cyclic groups (OD = 2 except for {1,2,4,6}), non-cyclic $p$-groups (OD = 3 for generalized quaternion, and OD = 4 otherwise), and a full nilpotent-group classification yielding OD = 3 or 4 under explicit group-theoretic conditions, with a polynomial-time algorithm to compute OD for nilpotent groups. The study also connects to enhanced power and commuting graphs, deriving bounds and highlighting avenues for further exploration. Overall, the results show how group structure informs precise diameter outcomes in related graph models and enable efficient computation of these diameters in nilpotent cases.

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.

On Oriented Diameter of Power Graphs

TL;DR

This work determines the oriented diameter of power graphs for broad classes of finite groups. It proves a sharp upper bound: any 2-edge-connected power graph Pow(G) has OD ≤ 4, and it provides precise OD values for cyclic groups (OD = 2 except for {1,2,4,6}), non-cyclic -groups (OD = 3 for generalized quaternion, and OD = 4 otherwise), and a full nilpotent-group classification yielding OD = 3 or 4 under explicit group-theoretic conditions, with a polynomial-time algorithm to compute OD for nilpotent groups. The study also connects to enhanced power and commuting graphs, deriving bounds and highlighting avenues for further exploration. Overall, the results show how group structure informs precise diameter outcomes in related graph models and enable efficient computation of these diameters in nilpotent cases.

Abstract

In this paper, we study the oriented diameter of power graphs of groups. We show that a -edge connected power graph of a finite group has oriented diameter at most . We prove that the power graph of the cyclic group of order has oriented diameter for all . For non-cyclic finite nilpotent groups, we show that the oriented diameter of corresponding power graphs is at least . Moreover, we provide necessary and sufficient conditions for the oriented diameter of -edge connected power graphs of finite non-cyclic nilpotent groups to be either or . 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.
Paper Structure (16 sections, 27 theorems, 10 figures)

This paper contains 16 sections, 27 theorems, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Oriented Diameter of Power Graphs
  4. Oriented Diameter of Power Graphs of Cyclic Groups
  5. Oriented Diameter of Power Graphs of $p$-groups
  6. Oriented Diameter of Power Graphs of Nilpotent Groups
  7. Oriented Diameter of Enhanced Power Graphs and Commuting Graphs
  8. Extended Preliminary
  9. Appendix
  10. Power graphs violating the conditions given by Czabarka et al. and Cochran et al.
  11. Power graph of $\mathbb{Z}_6$
  12. Proof of \ref{['direct product of co-prime order']}
  13. An observation for the proof of \ref{['nil: OD strictly greater than 3']} and \ref{['nil: one 2-order']}
  14. Examples of nilpotent groups corresponding to \ref{['nil: main result']}
  15. Proof of \ref{['nil: no mcs of order 2pk']}
  16. ...and 1 more sections

Key Result

Theorem 1

fomin2004complexity For every $n \geq 3$, $OD(K_n)=2$ except $n = 4$, and $OD(K_4)=3$. Moreover, for every $n \geq 5$, every strong orientation of $K_n$ with diameter $2$ can be extended to a strong orientation of $K_{n+1}$ with diameter $2$ and this extension can be constructed in linear time.

Figures (10)

  • Figure 1: An orientation of $K_4$ with $ecc(e)=2$
  • Figure 2: A partial orientation of $Pow(\mathbb{Z}_n)$ with diameter $3$.
  • Figure 3: The $P_4$-gadget
  • Figure 6: Choosing the directed path $c_{11}ec_{21}$ forces the directed edges shown in the figure.
  • Figure 7: A partial orientation $\mathcal{O}$ of $Pow(Q_{2^n})$ with diameter $3$. Here $y$ is the element of order $2$.
  • ...and 5 more figures

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 2
  • ...and 40 more