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arxiv: 2604.12404 · v1 · submitted 2026-04-14 · 🧮 math.CO

Maximizing the Steklov eigenvalues on trees with a diameter constraint

Jiangdong Ai , Huiqiu Lin , Yongtang Shi This is my paper

Pith reviewed 2026-05-10 14:56 UTC · model grok-4.3

classification 🧮 math.CO
keywords Steklov eigenvaluetreesdiameterspider treesmaximizationDirichlet-to-Neumann operatorboundary flux
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The pith

For trees of any odd diameter the first nonzero Steklov eigenvalue reaches its maximum exactly on a family of spider trees whose branch lengths are nearly equal and fixed by a counting rule on the number of leaves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper completes the geometric classification of maximizers for the first nonzero Steklov eigenvalue under a fixed diameter constraint. It proves that when the diameter is odd and at least five, the maximum is attained by spider trees whose branches have lengths that differ by at most one, with the precise distribution determined by the arithmetic relation between the total number of leaves and half the radius. The same family, called generalized almost seesaw trees, covers every odd diameter once the parameters are chosen accordingly. If the claim holds, the extremal trees are now known explicitly for every possible diameter, matching the even-diameter case already settled earlier. The argument proceeds by showing that any candidate tree can be reduced, without lowering the eigenvalue, first to a two-center form and then to a one-center spider whose optimal branch lengths solve a scalar root equation derived from the boundary flux quadratic form.

Core claim

For every odd diameter D = 2r + 1 at least 5 the largest possible value of the first nonzero Steklov eigenvalue on a tree of diameter D is realized precisely by the generalized almost seesaw trees AS(r, q + 2, c, t). These are spider trees whose branches have lengths differing by at most one; the parameters q, c, t are fixed by the position of the total number of vertices n relative to ceiling of r over 2. The proof establishes that no other tree of the same diameter can exceed this value and that the listed spiders achieve it.

What carries the argument

A reduction scheme that transforms an arbitrary tree first into a two-center profile and then into a one-center spider while preserving or increasing the Steklov eigenvalue, combined with a scalar root equation whose solutions give the optimal branch lengths for each one-center profile.

If this is right

  • The extremal trees for odd diameters are always spiders with at most two distinct branch lengths that differ by one.
  • The arithmetic rule that selects the parameters q, c, t depends only on the relation of n to ceiling of r over 2 and produces a unique family for each odd D.
  • Together with the even-diameter and diameter-three cases the classification is now complete for every diameter.
  • The inverse boundary quadratic form on fluxes supplies the variational tool that makes the reduction and the root equation work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique might classify maximizers for higher Steklov eigenvalues or for other boundary-value problems on trees.
  • Numerical checks on small odd diameters such as five and seven would give immediate verification of the arithmetic rule.
  • The boundary-flux viewpoint may extend to discrete or weighted trees where the classical distance-matrix methods do not apply directly.

Load-bearing premise

That any tree maximizing the eigenvalue can be reduced, without decreasing the value, to a one-center spider whose branch lengths satisfy the scalar root equation.

What would settle it

An explicit tree of odd diameter D whose first nonzero Steklov eigenvalue exceeds the value attained by the generalized almost seesaw tree AS(r, q + 2, c, t) constructed for the same D and the same number of vertices.

Figures

Figures reproduced from arXiv: 2604.12404 by Huiqiu Lin, Jiangdong Ai, Yongtang Shi.

Figure 1
Figure 1. Figure 1: The two explicit extremal candidates in Theorem 1.2 and Theorem 4.1, where [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

We study the first nonzero Steklov eigenvalue $\lambda_2(T,\delta\Omega)$ of the Dirichlet-to-Neumann operator on a finite tree $T$ with leaf boundary $\delta\Omega$, under a constraint on the diameter $D$. He and Hua [Calc. Var. PDE, 2022] showed that $\lambda_2(T) \leq 2/D$ for any tree of diameter $D$, with the even-diameter equality case fully characterized. For odd $D$, the geometric picture underlying the sharp configurations has remained unclear beyond diameter three. We determine this picture completely for all odd diameters $D = 2r+1 \geq 5$. The sharp value of $\lambda_2$ is achieved on spider trees with nearly-equidistributed branch lengths, forming the family of \emph{generalized almost seesaw trees} $\mathrm{AS}(r,q+2,c,t)$, prescribed by the arithmetic of $n$ relative to $\lceil r/2 \rceil$. Together with the results of He-Hua and Lin-Zhao [Bull. Lond. Math. Soc., 2025] for even diameters and diameter three, this completes the geometric classification for every diameter. The argument is based on a scalar root equation for one-center profiles, an inverse boundary quadratic form on boundary fluxes, and a reduction scheme from arbitrary trees to two-center profiles, and then to the one-center class. The inverse variational viewpoint may be regarded as a boundary analogue of the classical distance-matrix formalism for trees initiated by Graham and Lov\'asz [Adv. Math., 1978].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript determines the maximizers of the first nonzero Steklov eigenvalue λ₂ on trees of fixed odd diameter D=2r+1≥5. It proves that the sharp value is attained precisely on the family of generalized almost seesaw trees AS(r,q+2,c,t), which are spider trees whose branch lengths are as equal as possible subject to the diameter constraint; the precise parameters (q,c,t) are fixed by the arithmetic of the number of vertices n relative to ⌈r/2⌉. The argument proceeds by reducing an arbitrary tree first to a two-center profile and then to a one-center spider, solving a scalar root equation that identifies the unique maximizer within the one-center class, and using an inverse boundary quadratic form to control the variational characterization of λ₂. Together with the earlier results of He-Hua (even diameters) and Lin-Zhao (D=3), this yields a complete geometric classification for every diameter.

Significance. If the reduction and root-equation analysis are valid, the paper supplies the missing odd-diameter half of the classification, thereby completing a program initiated by He-Hua. The explicit construction of the extremal family AS(r,q+2,c,t) and the introduction of the inverse boundary quadratic form as a boundary analogue of the Graham-Lovász distance-matrix formalism constitute concrete advances. The work is self-contained once the prior even-diameter and D=3 cases are granted, and it furnishes falsifiable, explicitly describable extremal graphs.

minor comments (3)
  1. [§2.3] §2.3, Definition 2.7: the family AS(r,q+2,c,t) is introduced via a table of arithmetic cases; a single compact formula or pseudocode that generates (q,c,t) from (r,n) would make the statement of the main theorem easier to parse.
  2. [§4.2] §4.2, Lemma 4.5: the proof that the root equation has a unique positive solution for each admissible (r,q) is only sketched; supplying the monotonicity argument or the explicit derivative would strengthen the one-center analysis.
  3. [Figure 3] Figure 3: the three example trees for r=3 are drawn at different scales; uniform scaling and explicit labeling of the branch lengths would improve visual comparison with the claimed equidistribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our work, which correctly identifies the completion of the geometric classification of maximizers for λ₂ on trees of odd diameter D=2r+1≥5 via the family of generalized almost seesaw trees AS(r,q+2,c,t). We appreciate the recognition of the inverse boundary quadratic form and the reduction scheme as concrete advances. The report recommends minor revision but lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central derivation for odd diameters D=2r+1≥5 proceeds via an independent reduction scheme from arbitrary trees to two-center profiles then one-center spiders, a scalar root equation identifying maximizers within the one-center class, and an inverse boundary quadratic form on fluxes. These steps are explicitly distinguished from the cited He-Hua and Lin-Zhao results, which apply only to even diameters and D=3 and are invoked solely to complete the overall classification. No load-bearing self-citation, self-definitional equivalence, fitted input renamed as prediction, or ansatz smuggling occurs in the odd-diameter argument; the variational characterization and diameter constraint are preserved throughout without circular reduction to prior inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the correctness of the reduction from arbitrary trees to two-center and then one-center profiles, the existence and uniqueness properties of the scalar root equation, and the prior classification results for even diameters and D=3. No free parameters are introduced; the family AS(r,q+2,c,t) is defined combinatorially from the diameter and leaf count.

axioms (2)
  • domain assumption The reduction scheme maps any tree to an equivalent two-center profile without decreasing the eigenvalue.
    Invoked in the argument that reduces arbitrary trees to the one-center class.
  • domain assumption The scalar root equation for one-center profiles has a unique positive solution that corresponds to the maximizer.
    Used to locate the optimal branch lengths in the spider trees.
invented entities (1)
  • generalized almost seesaw trees AS(r,q+2,c,t) no independent evidence
    purpose: Explicit family of spider trees that achieve the sharp bound for odd diameters.
    New combinatorial object defined by the arithmetic of n relative to ceiling of r/2; no independent evidence supplied beyond the proof that they attain the bound.

pith-pipeline@v0.9.0 · 5594 in / 1515 out tokens · 125516 ms · 2026-05-10T14:56:27.634666+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    J. Ai, Y. Ji, X. Lian and K. Yang,Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf-boundary trees, Preprint, arXiv:2603.26251, 2026

    work page arXiv 2026

  2. [2]

    Al Sayed, B

    A. Al Sayed, B. Bogosel, A. Henrot, and F. Nacry,Maximization of the Steklov eigenvalues with a diameter constraint, SIAM J. Math. Anal. 53 (2021), no. 1, 710–729

    work page 2021

  3. [3]

    Colbois, A

    B. Colbois, A. Girouard, C. Gordon and D. A. Sher,Some recent developments on the Steklov eigenvalue problem, Rev. Mat. Complut.37(2024), 1–161. 21

    work page 2024

  4. [4]

    Girouard and I

    A. Girouard and I. Polterovich,Spectral geometry of the Steklov problem, J. Spectr. Theory 7(2017), no. 2, 321–359

    work page 2017

  5. [5]

    R. L. Graham and L. Lovász,Distance matrix polynomials of trees, Adv. Math.29(1978), no. 1, 60–88

    work page 1978

  6. [6]

    R. L. Graham and H. O. Pollak,On the addressing problem for loop switching, Bell System Tech. J.50(1971), 2495–2519

    work page 1971

  7. [7]

    Han and B

    W. Han and B. Hua,Steklov eigenvalue problem on subgraphs of integer lattices, Comm. Anal. Geom.31(2023), no. 2, 343–366

    work page 2023

  8. [8]

    Hassannezhad and L

    A. Hassannezhad and L. Miclo,Higher order Cheeger inequalities for Steklov eigenvalues, Ann. Sci. Éc. Norm. Supér. (4)53(2020), no. 1, 43–88

    work page 2020

  9. [9]

    He and B

    Z. He and B. Hua,Upper bounds for the Steklov eigenvalues on trees, Calc. Var. Partial Differential Equations61(2022), no. 3, Paper No. 101

    work page 2022

  10. [10]

    B. Hua, Y. Huang and Z. Wang,First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs, Calc. Var. Partial Differential Equations56(2017), no. 6, Paper No. 178

    work page 2017

  11. [11]

    H. Lin, L. Liu, Z. You and D. Zhao,Upper bounds of Steklov eigenvalues on graphs, J. Geom. Anal.36(2026), no. 3, Paper No. 95, 24 pp

    work page 2026

  12. [12]

    Lin and D

    H. Lin and D. Zhao,Maximize the Steklov eigenvalue of trees, Bull. Lond. Math. Soc.57 (2025), no. 12, 4149–4162

    work page 2025

  13. [13]

    Lin and D

    H. Lin and D. Zhao,The first Steklov eigenvalue of planar graphs and beyond, J. Lond. Math. Soc. (2)112(2025), no. 1, e70238

    work page 2025

  14. [14]

    Merris,The distance spectrum of a tree, J

    R. Merris,The distance spectrum of a tree, J. Graph Theory14(1990), no. 3, 365–369

    work page 1990

  15. [15]

    Perrin,Lower bounds for the first eigenvalue of the Steklov problem on graphs, Calc

    H. Perrin,Lower bounds for the first eigenvalue of the Steklov problem on graphs, Calc. Var. Partial Differential Equations58(2019), no. 2, Paper No. 67

    work page 2019

  16. [16]

    Perrin,Isoperimetric upper bound for the first eigenvalue of discrete Steklov problems, J

    H. Perrin,Isoperimetric upper bound for the first eigenvalue of discrete Steklov problems, J. Geom. Anal.31(2021), no. 8, 8144–8155

    work page 2021

  17. [17]

    Shi and C

    Y. Shi and C. Yu,Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs, Proc. Amer. Math. Soc.150(2022), no. 4, 1505–1517

    work page 2022

  18. [18]

    Shi and C

    Y. Shi and C. Yu,A Lichnerowicz-type estimate for Steklov eigenvalues on graphs and its rigidity, Calc. Var. Partial Differential Equations61(2022), no. 3, Paper No. 98

    work page 2022

  19. [19]

    Stekloff,Sur les problèmes fondamentaux de la physique mathématique, Ann

    W. Stekloff,Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. Ècole Norm. Sup. (3)19(1902), 191–259

    work page 1902

  20. [20]

    Tschanz,Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs, Ann

    L. Tschanz,Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs, Ann. Glob. Anal. Geom.61(2022), no. 1, 37–55

    work page 2022

  21. [21]

    Yu and Y

    C. Yu and Y. Yu,Monotonicity of Steklov eigenvalues on graphs and applications, Calc. Var. Partial Differential Equations63(2024), Paper No. 79. 22

    work page 2024