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<p>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 <i>n</i> has oriented diameter 2 for all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ne 1,2,4,6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≠</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that the oriented diameter of 2-edge connected power graphs of non-cyclic nilpotent groups is either 3 or 4. Moreover, we provide necessary and sufficient conditions to determine when such graphs have oriented diameter 3 and when these graphs have diameter 4. This, in turn, gives a polynomial time algorithm for computing the oriented diameter of the power graph of a given nilpotent group.</p>

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On Oriented Diameter of Power Graphs

  • 期刊论文

  • 发表日期：

    2026年2月9日

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  • Deepu Benson,
  • Bireswar Das,
  • Dipan Dey,
  • Jinia Ghosh

摘要

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 \(n\ne 1,2,4,6\) n ≠ 1 , 2 , 4 , 6 . We show that the oriented diameter of 2-edge connected power graphs of non-cyclic nilpotent groups is either 3 or 4. Moreover, we provide necessary and sufficient conditions to determine when such graphs have oriented diameter 3 and when these graphs have diameter 4. This, in turn, gives a polynomial time algorithm for computing the oriented diameter of the power graph of a given nilpotent group.