Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Graph Eigenvalues and Projection Constants

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every graph's k-th largest adjacency eigenvalue is bounded above by a universal constant times the number of vertices, and the constant is sharp for k = 2, 3, 4, 8, and 24.

desk verdict A genuinely new reduction from graph eigenvalues to projection constants, proving c3=1/3 and matching upper bounds for k=4,8,24; the upper bound is solid, tightness for 8 and 24 leans on external frame constructions. read the letter →

arxiv 2608.02429 v1 pith:EAEEXEHD submitted 2026-08-03 math.CO

Varun Sivashankar , Quanyu Tang , Tanay Wakhare This is my paper · ORCID
classification math.CO MSC 05C5015A1846B2052C35
keywords grapheigenvaluesadjacencymatrixabsoluteprojectionconstantorthogonalprojectionsequiangularlinesregulartwo-graphsextremaleigenvalueboundspositivedefinitekernels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a universal upper bound on the k-th largest adjacency eigenvalue λ_k(G) of any graph on n vertices: λ_k(G) ≤ ((k−2)√(k+1)+2)/(2k(k−1)) n − 1. The bound is tight for k = 2, 3, 4, 8, and 24, settling the last open small-k case c_3 = 1/3 and giving exact extremal constants for the other tight values. The proof reduces the graph-eigenvalue problem to an extremal problem for orthogonal projections and combines this with the sharpest known bound on the absolute projection constant in dimension k−1. A self-contained analytic proof of that projection bound is given, via positive kernels on the sphere, and it yields a strict improvement in even dimensions r ≥ 4 for which r+2 is not a perfect square. The same machinery also resolves a conjecture on the limiting normalized sum of the k largest eigenvalues.

What carries the argument

The load-bearing object is the absolute projection constant γ(r), defined as the supremum over all rank-r orthogonal projection matrices Q of the average of the absolute values of their entries, γ(r) = sup_N (1/N) max_{Q∈P_r(N)} ∥Q∥_1. The graph-to-projection step uses the variational (min-max) principle for sums of eigenvalues to write the sum of the r smallest eigenvalues as a minimum of tr(AQ) over rank-r projections, bounding it below by −γ(r)n/2; complemented with an interlacing inequality for the eigenvalues of a matrix sum applied to G and its complement, this yields λ_k(G) ≤ γ(k−1)/(2(k−1)) n − 1. The analytic estimate for γ(r) uses two positive-semidefinite kernels on the sphere (de

What would settle it

Compute, for a single large n, the largest λ_4/n among all graphs on n vertices; if any value exceeds ((2)√5+2)/(24) − 1/n ≈ 0.2057 − 1/n, the main theorem fails. Alternatively, for the projection side, numerically maximize ∥Q∥_1/n over rank-3 orthogonal projections; exceeding (3+√5)/(1+√5) ≈ 1.309 would disprove the bound. A feasible target: check whether the known explicit weighted cores give γ(4) > 1.8500; if a rank-4 projection with ∥Q∥_1/n > 1.8500 exists, the projection bound for r=4 is violated.

Watch

Extended reading notes

Core claim

The central claim is a reduction: for r = k−1, the extremal value of λ_k(G)/n is dominated by the absolute projection constant γ(r), via the inequality λ_k(G) ≤ γ(r)/(2r) n − 1. Combining this with the projection-constant bound γ(r) ≤ (r+√(r+2))/(1+√(r+2)) gives the explicit coefficient α_k. Tightness for k ∈ {2,3,4,8,24} is achieved by graphs built from equiangular line systems with common angle 1/√(r+2): the extremal sign matrices and regular two-graph constructions match the bound exactly. For k = 3 this yields the sharp value c_3 = 1/3; for k = 4, 8, and 24 it yields α_4 = (1+√5)/12, α_8 = 5/28, and α_24 = 7/69, agreeing with previously constructed lower bounds.

Load-bearing premise

The tightness claim for k = 8 and 24 depends on two externally supplied facts: that maximal equiangular line systems in those dimensions exist, and that the associated regular two-graph has the exact spectrum used; if either fact fails, the equality cases for those k collapse, though the upper bound itself is proven independently.

Editorial extensions

If this is right

  • For k=3, the sharp constant becomes c_3 = 1/3: every graph satisfies λ_3(G) ≤ n/3 − 1, and the bound is attained in the limit.
  • For k=4, 8, and 24, the extremal constants are respectively (1+√5)/12, 5/28, and 7/69, matching all known lower-bound constructions and proving optimality.
  • The limiting value of the maximal normalized sum of the k largest adjacency eigenvalues, τ_k, satisfies an explicit strict inequality below the previous general bound, settling the open conjecture with an explicit ε_k.
  • For even r ≥ 4 with r+2 not a perfect square, the absolute projection constant γ(r) is strictly smaller than the main general bound, by an explicit (albeit tiny) quantity.
  • The reduction implies that the multiplicity of the second eigenvalue of a connected non-complete graph is at most O((n/(λ_2+1))^2), improving the previous order bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The tight cases k = 4, 8, and 24 line up with the dimensions where maximal real equiangular tight frames exist, which are the same exceptional dimensions behind optimal sphere packings; the paper's constants therefore suggest a structural connection between extremal graph spectra and exceptional root/lattice geometries.
  • The identity λ_k(G) ≤ γ(k−1)/(2(k−1)) n − 1 suggests the two extremal problems — graph eigenvalues and projection constants — may be strictly separated for some k, since the numerical weighted-core constructions in the paper produce slightly different lower bounds for the two sides; resolving whether c_{r+1} < γ(r)/(2r) for some r is a natural next step.
  • The proof route via positive-semidefinite kernels and scalar majorants is general enough that refining the kernels (e.g., degree-6 or higher) could yield further improvements to the projection constant and hence to graph eigenvalue bounds in dimensions not covered by the strict-improvement theorem.
  • Since the graph bound uses only the coarse information that off-diagonal entries lie in [0,1], a natural extension is to blend this dense-graph method with sparsity or degree information to get bounds that degrade gracefully for sparse graphs.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves a general upper bound on the k-th largest adjacency eigenvalue of an n-vertex graph: λ_k(G) ≤ γ(k−1)/(2(k−1)) n − 1 ≤ α_k n − 1, where α_k = ((k−2)√(k+1)+2)/(2k(k−1)). The proof reduces the graph problem to a bound on the absolute projection constant γ(r), then gives a self-contained Gegenbauer-polynomial proof of the Deręgowska–Lewandowska projection bound, an improved bound in even nonsquare dimensions, a resolution of Nikiforov's conjecture on sums of the k largest eigenvalues, and numerical weighted-core lower bounds. Tightness is claimed for k ∈ {2,3,4,8,24}, with k=8 and k=24 relying on maximal real equiangular tight frames in R^7 and R^23.

Significance. The upper-bound chain is elegant and largely self-contained. It settles the last open small-k case c_3 = 1/3, gives c_4 = (1+√5)/12, and yields the first general 1/√k-type bound. Proposition 6.1 resolves Nikiforov's Conjecture 4.2 with an explicit ε_k, and the slack identity in Lemma 4.4 is a transparent, parameter-free derivation. The paper also supplies reproducible ancillary code for the numerical lower bounds. If the tightness dependencies are supplied, this would be a substantial contribution to the extremal theory of graph eigenvalues.

major comments (2)
  1. [Section 5.1, paragraph 'The extremal equiangular line systems...'] Tightness for k=8 and k=24, i.e. the equalities c_8 = 5/28 and c_24 = 7/69, depends on the existence of maximal real equiangular tight frames with N=(r+1)r/2 lines in R^r for r=7 and r=23. This is cited only to [Gil18], an arXiv preprint, with no construction or standard peer-reviewed reference. This is a load-bearing external input: without these frames the equality part of Theorem 2.1 collapses, leaving only the upper bound. Please include explicit line-system constructions (for example from the E8 and Leech lattices) or replace [Gil18] with a standard reference such as [LS73] or [Sei76] and outline the construction in the text.
  2. [Section 5.1, 'The spectrum of this graph is well understood'] The identity λ_{r+1}(G0) = β_r/(2r)|V(G0)| − 1 is imported from [Sei76, GH92] without proof. This identity is used in all tight cases, including the critical k=8,24 cases, so it should be derived in the paper rather than asserted by citation. The block form A(G0)+I = (1/2)[[J+B,J−B],[J−B,J+B]] makes this a short derivation from the already-computed spectrum of B; adding it would make the lower-bound argument self-contained.
minor comments (4)
  1. [Proof of Theorem 2.1] After substituting s = √(k+1), the displayed inequality γ(k−1)/(2(k−1)) ≤ ... is actually an equality: (k−1+s)/(2(k−1)(1+s)) = ((k−2)s+2)/(2k(k−1)). Showing this one-line rationalization would clarify the transition to α_k.
  2. [Lemma 4.3] The lemma states ∥Q∥_1 ≤ a r n, where r is the rank and a is the scalar majorant coefficient. This is correct but easy to misread as ∥Q∥_1 ≤ a n. Consider writing the bound as ∥Q∥_1 ≤ (a r) n and noting that the final projection-constant bound is β_r = r a_r.
  3. [Table 1 and Section 5.3] The table would benefit from explicit column headers in the printed text, and the reader should be told which rows are verified by the ancillary script versus merely heuristic search output. The current statement is clear that the searches are heuristic, but the distinction between verified construction data and exploratory lower-bound evidence should be made even more explicit.
  4. [Appendix C] The 11×11 matrix B for r=4 is defined via the displayed matrices R, v, and the weight pattern, but the typesetting makes the block structure hard to read. Since the full data is in the ancillary files, a simpler description or a reference to the JSON file would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the graph-eigenvalue upper bound is derived in-paper from a reproof of the projection-constant bound, and the tightness cases depend only on external classical constructions, not on self-citation or fitted inputs.

full rationale

The derivation chain is self-contained at every load-bearing point. Theorem 2.3 is proven directly from Ky Fan's principle and the definition of γ(r); Theorem 2.2 combines it with a Weyl inequality; the bound on γ(r) used in Theorem 2.1 is Theorem 4.1, which the paper reproves in Section 4 without importing the [DL23] proof as a black box. That reproof rests on Gram-matrix kernels f2^r and f4^r (Lemmas 3.2–3.3), Lemma 4.2's inequality C²+rX²≤rn, and Lemma 4.4's explicit factorization (1−t)(st−1)²(st+s+2)/(2(s+1)²), followed by a nonnegativity argument. The contact points t=1 and t=1/√(r+2) in Remark 4.5 are heuristics for selecting coefficients, not fitted data: the polynomial inequality is then verified for every t∈[−1,1], so the bound is not equal to its motivation by construction. The tightness half for k=4,8,24 cites the existence of maximal real equiangular tight frames in R^7 and R^23 ([Gil18]) and the regular two-graph spectral identity ([Sei76, GH92]); these are external classical facts rather than self-citations, and although their omission makes the equality cases less self-contained, a defect there would be a correctness/completeness issue, not circularity. Table 1 is explicitly heuristic and is presented as lower-bound evidence, not as a prediction. No load-bearing step reduces to the result being proved, and no self-citation carries the argument. Score 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The main theorem is parameter-free: α_k follows from exact factorizations and the definition of γ(k−1), and the DL23 projection bound is re-proven in-paper rather than imported. Free parameters enter only in secondary numerical lower bounds (Table 1 cores) and in hand-chosen stability thresholds whose values set the (crude) size of the strict improvement. The tightness claims import two external facts: equiangular-frame existence and the two-graph spectral identity.

free parameters (3)
  • r=4 core weights (a_Q, b_Q, c_Q) = a≈0.0978456601, b≈0.1072573795, c≈0.0549886699
    Numerical optimization outputs for the Table 1 lower-bound construction; the resulting Φ_B(p) ≈ 1.8500840104 is a directly verifiable lower bound, not an analytic derivation.
  • Stability thresholds η_r, e, and the defect threshold e²/(128r) = η_r = 1/(r(2r)^{2r+1}), e = η_r/r
    Hand-chosen to make Appendix B's slack algebra work; they set the (extremely crude) magnitude of the strict improvement in Theorem 4.6. Outputs of proof design, not of data.
  • Gegenbauer contact coefficients (a_r, b_r, ρ_r) = a_r=(s²+s−2)/(r(s+1)), b_r=s(s²+2s+3)/(2(s+1)²), ρ_r=s³/(2(s+1)²), s=√(r+2)
    Determined by forcing contact at t=1 and t=1/s (Remark 4.5); an ansatz, but subsequently verified by an exact factorization identity, so the bound holds for all t and r ≥ 2. Not fitted to data.
assumptions (6)
  • domain assumption Existence of maximal real equiangular tight frames in R^r for r ∈ {1,2,3,7,23} (r = 7, 23 from E8 and Leech lattices)
    Needed for the tightness claims k = 4, 8, 24 and for matching lower bounds; imported from [Gil18, DGS77, LS73] without proof in §5.1.
  • domain assumption Regular two-graph spectral identity: λ_{r+1}(G0) = β_r/(2r)|V(G0)| − 1
    Quoted from [Sei76, GH92] in §5.1; this single identity carries all five tightness cases.
  • standard math Ky Fan min/max principle and Weyl's additive eigenvalue inequality
    Workhorses of Theorems 2.3 and 2.2 (cited to [HJ12, Cor. 4.3.39] and [HJ12, §4.3]).
  • standard math Motzkin–Straus weighted Turán theorem
    Lemma B.6 converts the good-pair density bound g > 1 − 1/r into a K_{r+1} clique in the contradiction argument for Theorem 4.6.
  • standard math Direct Gram-matrix positivity of the rank-two and rank-four kernels f_2^r, f_4^r
    Proven in Lemmas 3.2, 3.3 and Appendix A via tensor Gram representations; Schoenberg's theorem [Sch42] is cited but not actually required.
  • standard math Seidel characteristic polynomial modulo 2: χ_S(x) ≡ x(x+1)^{m−1} for odd m
    Lemma B.2; the linchpin of the non-square obstruction in Lemma B.3 that forces the strict improvement in Theorem 4.6.
invented entities (1)
  • Finite weighted core (B, p) independent evidence
    purpose: Finite-type reduction for lower bounds on γ(r) and c_{r+1} (Proposition 5.1) and for the Table 1 constructions
    A construction device, not a postulated physical entity: each core is a concrete (±1 sign matrix, weight vector) whose objective Φ_B(p) = Σ λ_j(K_p) is directly recomputable, and the paper promises a verification script. Independent falsifiable handle = recomputation from (B, p).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Graph Eigenvalues and Projection Constants." pith.science (2026). https://pith.science/paper/EAEEXEHD

@misc{pith2026260802429,
  author       = {Pith},
  title        = {Pith review of: Graph Eigenvalues and Projection Constants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAEEXEHD}},
  note         = {Machine review of arXiv:2608.02429}
}
abstract

For an integer $k\ge2$, let $\lambda_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $\gamma(r)$ due to Der\k{e}gowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of K\"onig and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples

    math.FA 2026-08 conditional novelty 6.0 of 10

    The paper shows biangular tight frames make two projection constants coincide, derives a new lower bound from simplex edge midpoints, and conjectures exact maximal projection constants in dimensions 6 and 8.

Reference graph

Works this paper leans on

36 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [1]

    and Goethals, J

    Delsarte, P. and Goethals, J. M. and Seidel, J. J. , title =. Geometriae Dedicata , volume =. 1977 , doi =

    1977

  2. [2]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , title =

  3. [3]

    , title =

    Hong, Y. , title =. Acta Mathematicae Applicatae Sinica , volume =. 1988 , doi =

    1988

  4. [4]

    , title =

    Hong, Y. , title =. Discrete Mathematics , volume =. 1993 , doi =

    1993

  5. [5]

    and Li, S

    Leonida, G. and Li, S. , title =. Linear Algebra and its Applications , volume =. 2026 , doi =

    2026

  6. [6]

    Strengthened upper bound on the third eigenvalue of graphs

    Li, S. , title =. arXiv preprint arXiv:2501.07494 , year =. doi:10.48550/arXiv.2501.07494 , eprint =

    work page Pith review arXiv doi:10.48550/arxiv.2501.07494

  7. [7]

    , title =

    Linz, W. , title =. Graphs and Combinatorics , volume =. 2023 , doi =

    2023

  8. [8]

    , title =

    Nikiforov, V. , title =. Linear Algebra and its Applications , volume =. 2015 , doi =

    2015

Show all 36 references
  1. [9]

    , title =

    Mohar, B. , title =. Journal of Combinatorial Theory, Series B , volume =. 2009 , doi =

    2009

  2. [10]

    Ebrahimi, J. B. and Mohar, B. and Nikiforov, V. and Ahmady, A. S. , title =. Linear Algebra and its Applications , volume =. 2008 , doi =

    2008

  3. [11]

    and Liu, L

    Kumar, H. and Liu, L. and Monterde, H. and Pragada, S. and Tait, M. , title =. arXiv preprint arXiv:2604.00512 , year =. doi:10.48550/arXiv.2604.00512 , eprint =

    doi arXiv

  4. [12]

    and Antley, C

    Agarwal, S. and Antley, C. and Aulenbacher, J. and Brooks, G. and Gonzalez, I. and Hawranick, L. and Linz, W. and Lu, L. and Williams, A. , title =. arXiv preprint arXiv:2607.15941 , year =. doi:10.48550/arXiv.2607.15941 , eprint =

    doi arXiv

  5. [13]

    Powers, D. L. , title =. Linear Algebra and its Applications , volume =. 1989 , doi =

    1989

  6. [14]

    Schoenberg, I. J. , title =. Duke Mathematical Journal , volume =. 1942 , doi =

    1942

  7. [15]

    Journal of Algebra , volume=

    Equiangular lines , author =. Journal of Algebra , volume=. 1973 , publisher=

    1973

  8. [16]

    Seidel, J. J. , title =. Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I , series =. 1976 , publisher =

    1973

  9. [17]

    Godsil, C. D. and Hensel, A. D. , title =. Journal of Combinatorial Theory, Series B , volume =. 1992 , doi =

    1992

  10. [18]

    Annals of mathematics , pages=

    The sphere packing problem in dimension 8 , author =. Annals of mathematics , pages=. 2017 , publisher=

    2017

  11. [19]

    Annals of mathematics , volume=

    The sphere packing problem in dimension 24 , author =. Annals of mathematics , volume=. 2017 , publisher=

    2017

  12. [20]

    arXiv preprint arXiv:1809.05739 , year=

    Equiangular lines, incoherent sets and quasi-symmetric designs , author =. arXiv preprint arXiv:1809.05739 , year=

    arXiv

  13. [21]

    1971 , publisher=

    Introduction to Fourier analysis on Euclidean spaces , author =. 1971 , publisher=

    1971

  14. [22]

    Taylor, D. E. , title =. Proceedings of the London Mathematical Society , series =. 1977 , doi =

    1977

  15. [23]

    Proceedings of the London Mathematical Society , volume=

    The difference between consecutive primes, II , author =. Proceedings of the London Mathematical Society , volume=. 2001 , publisher=

    2001

  16. [24]

    Projection constants , journal =

    Gr. Projection constants , journal =. 1960 , doi =

    1960

  17. [25]

    Chalmers, B. L. and Lewicki, G. , title =. Journal of Functional Analysis , volume =. 2009 , doi =

    2009

  18. [26]

    Chalmers, B. L. and Lewicki, G. , title =. Studia Mathematica , volume =. 2010 , doi =

    2010

  19. [27]

    , title =

    Basso, G. , title =. Journal of Functional Analysis , volume =. 2019 , doi =

    2019

  20. [28]

    and Cox, C

    Bukh, B. and Cox, C. , title =. Israel Journal of Mathematics , volume =. 2020 , doi =

    2020

  21. [29]

    Motzkin, T. S. and Straus, E. G. , title =. Canadian Journal of Mathematics , volume =. 1965 , doi =

    1965

  22. [30]

    Talbot, J. M. , title =. Combinatorics, Probability and Computing , volume =. 2002 , doi =

    2002

  23. [31]

    Norms of minimal projections , journal =

    K. Norms of minimal projections , journal =. 1994 , doi =

    1994

  24. [32]

    Spaces with maximal projection constants , journal =

    K. Spaces with maximal projection constants , journal =. 2003 , doi =

    2003

  25. [33]

    , title =

    Kobos, T. , title =. 2025 , eprint =

    2025

  26. [34]

    and Skrzypek, L

    Foucart, S. and Skrzypek, L. , title =. Journal of Mathematical Analysis and Applications , volume =. 2017 , doi =

    2017

  27. [35]

    A simple proof of the

    Der. A simple proof of the. Journal of Functional Analysis , volume =. 2023 , doi =

    2023

  28. [36]

    On the value of the fifth maximal projection constant , journal =

    Der. On the value of the fifth maximal projection constant , journal =. 2022 , doi =

    2022

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.