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<p>The algebraic connectivity of <i>G</i> is the second smallest eigenvalue of its Laplacian matrix. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathscr {T}(n, d) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the set of all trees with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( n \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> vertices and diameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( d \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>d</mi> </math></EquationSource> </InlineEquation>. In this paper, we identify the trees in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathscr {T}(n, d)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that achieve the second largest algebraic connectivity when <i>d</i> is odd. If <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( d \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>d</mi> </math></EquationSource> </InlineEquation> is even, a conjecture about which trees achieve the second largest algebraic connectivity is also proposed.</p>

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The second largest algebraic connectivity of trees with fixed diameter

  • 期刊论文

  • 发表日期：

    2026年3月13日

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  • Yi-Chen Yang,
  • Ji-Ming Guo

摘要

The algebraic connectivity of G is the second smallest eigenvalue of its Laplacian matrix. Let \( \mathscr {T}(n, d) \) T ( n , d ) denote the set of all trees with \( n \) n vertices and diameter \( d \) d . In this paper, we identify the trees in \(\mathscr {T}(n, d)\) T ( n , d ) that achieve the second largest algebraic connectivity when d is odd. If \( d \) d is even, a conjecture about which trees achieve the second largest algebraic connectivity is also proposed.