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<p>In this paper, we solve a minimum problem for Steklov eigenvalues on combinatorial graphs that is an analogue of the minimum problem solved by (Friedman in Duke Math. J. 83(1): 1–18 1996) for Laplacian eigenvalues by extending Friedman’s theory (Friedman in Duke Math. J. 69(3): 487–525 1993) of nodal domains for Laplacian eigenfunctions to Steklov eigenfunctions. More precisely, we mainly show that the minimum of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(i^\textrm{th}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>i</mi> <mtext>th</mtext> </msup> </math></EquationSource> </InlineEquation> Steklov eigenvalue on a connected combinatorial graph with <i>n</i> vertices is essentially attained by a star with each arm a minimal broom when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(i\not \mid n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, and attained by a regular comb with each tooth a minimal broom when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(i\mid n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∣</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Minimal Steklov Eigenvalues on Combinatorial Graphs

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Chengjie Yu,
  • Yingtao Yu

摘要

In this paper, we solve a minimum problem for Steklov eigenvalues on combinatorial graphs that is an analogue of the minimum problem solved by (Friedman in Duke Math. J. 83(1): 1–18 1996) for Laplacian eigenvalues by extending Friedman’s theory (Friedman in Duke Math. J. 69(3): 487–525 1993) of nodal domains for Laplacian eigenfunctions to Steklov eigenfunctions. More precisely, we mainly show that the minimum of the \(i^\textrm{th}\) i th Steklov eigenvalue on a connected combinatorial graph with n vertices is essentially attained by a star with each arm a minimal broom when \(i\not \mid n\) i ∤ n , and attained by a regular comb with each tooth a minimal broom when \(i\mid n\) i ∣ n .