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<p>Given a finite group <i>G</i>, the difference graph of <i>G</i>, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {D}(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, is the difference of the enhanced power graph of <i>G</i> and the power graph of <i>G</i>, with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of 4. Furthermore, we generalize and refine the result by Biswas et al. (2024) by classifying all nilpotent groups whose difference graph has diameter <i>k</i>, for each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\le 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The diameter and dominating sets of the difference graph of a nilpotent group

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Xuanlong Ma,
  • Samir Zahirović,
  • Katarina Žigerović

摘要

Given a finite group G, the difference graph of G, denoted by \(\mathcal {D}(G)\) D ( G ) , is the difference of the enhanced power graph of G and the power graph of G, with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of 4. Furthermore, we generalize and refine the result by Biswas et al. (2024) by classifying all nilpotent groups whose difference graph has diameter k, for each \(k\le 4\) k ≤ 4 .