The Steklov eigenvalue problem is a classical eigenvalue problem in spectral geometry. In this paper, we study the first (non-trivial) Steklov eigenvalue <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> of graph G with boundary B. Let D be the maximum vertex degree. Using metrical deformation via flows, we first show that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma _2 = \mathcal {O}\left( \frac{D g^3}{|B|}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mn>2</mn> </msub> <mo>=</mo> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <mfrac> <mrow> <mi>D</mi> <msup> <mi>g</mi> <mn>3</mn> </msup> </mrow> <mrow> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for graphs of orientable genus <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|B| \ge \max \{3 \sqrt{g},|V|^{\frac{1}{4} + \epsilon }, (9 \sqrt{g})^{\frac{1}{2}+\frac{1}{8\epsilon }}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> <mo>≥</mo> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <mn>3</mn> <msqrt> <mi>g</mi> </msqrt> <mo>,</mo> <mo stretchy="false">|</mo> <mi>V</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mi>ϵ</mi> </mrow> </msup> <mo>,</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>9</mn> <msqrt> <mi>g</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>ϵ</mi> </mrow> </mfrac> </mrow> </msup> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\epsilon > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. This can be seen as a discrete analogue of Karpukhin’s bound. Secondly, we prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sigma _2 \le \frac{8D+4X}{|B|}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mn>2</mn> </msub> <mo>≤</mo> <mfrac> <mrow> <mn>8</mn> <mi>D</mi> <mo>+</mo> <mn>4</mn> <mi>X</mi> </mrow> <mrow> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> based on planar crossing number X. Thirdly, we show that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma _2 \le \frac{|B|}{|B|-1} \cdot \delta _B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mn>2</mn> </msub> <mo>≤</mo> <mfrac> <mrow> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>·</mo> <msub> <mi>δ</mi> <mi>B</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\delta _B$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>B</mi> </msub> </math></EquationSource> </InlineEquation> denotes the minimum degree for boundary vertices in B. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues.

Upper Bounds of Steklov Eigenvalues on Graphs

期刊论文
发表日期：

2026年1月30日

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Huiqiu Lin,
Lianping Liu,
Zhe You,
Da Zhao

摘要

The Steklov eigenvalue problem is a classical eigenvalue problem in spectral geometry. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma _2$ $σ_{2}$ of graph G with boundary B. Let D be the maximum vertex degree. Using metrical deformation via flows, we first show that $\sigma _2 = \mathcal {O}\left( \frac{D g^3}{|B|}\right) $ $σ_{2} = O (\frac{D g^{3}}{| B |})$ for graphs of orientable genus $g > 0$ $g > 0$ if $|B| \ge \max \{3 \sqrt{g},|V|^{\frac{1}{4} + \epsilon }, (9 \sqrt{g})^{\frac{1}{2}+\frac{1}{8\epsilon }}\}$ $| B | \geq max {3 \sqrt{g}, | V |^{\frac{1}{4} + ϵ}, {(9 \sqrt{g})}^{\frac{1}{2} + \frac{1}{8 ϵ}}}$ for some $\epsilon > 0$ $ϵ > 0$ . This can be seen as a discrete analogue of Karpukhin’s bound. Secondly, we prove that $\sigma _2 \le \frac{8D+4X}{|B|}$ $σ_{2} \leq \frac{8 D + 4 X}{| B |}$ based on planar crossing number X. Thirdly, we show that $\sigma _2 \le \frac{|B|}{|B|-1} \cdot \delta _B$ $σ_{2} \leq \frac{| B |}{| B | - 1} \cdot δ_{B}$ , where $\delta _B$ $δ_{B}$ denotes the minimum degree for boundary vertices in B. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues.