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<p>A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(r \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k \ge r + 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist infinitely many nut graphs with <i>r</i> vertex orbits and <i>k</i> edge orbits. Here, we extend this result by finding all the pairs (<i>r</i>,&#xa0;<i>k</i>) for which there exists a nut graph with <i>r</i> vertex orbits and <i>k</i> edge orbits. In particular, we show that for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, there are infinitely many Cayley nut graphs with <i>k</i> edge orbits and <i>k</i> arc orbits.</p>

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Nut graphs with a prescribed number of vertex and edge orbits

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月8日

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  • Nino Bašić,
  • Ivan Damnjanović

摘要

A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even \(r \ge 2\) r ≥ 2 and any \(k \ge r + 1\) k ≥ r + 1 , there exist infinitely many nut graphs with r vertex orbits and k edge orbits. Here, we extend this result by finding all the pairs (r, k) for which there exists a nut graph with r vertex orbits and k edge orbits. In particular, we show that for any \(k \ge 2\) k ≥ 2 , there are infinitely many Cayley nut graphs with k edge orbits and k arc orbits.