{"id":"a42f5274-26b0-481d-ab9e-da01a21f7ee0","arxiv_id":"2608.02429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"λ_k(G) ≤ ((k−2)√(k+1)+2)n/(2k(k−1)) − 1 for all graphs, tight for k ∈ {2,3,4,8,24}, resolving c₃ = 1/3 and Nikiforov's Conjecture 4.2.","lead":"For every graph on n vertices, this paper bounds the k-th largest adjacency eigenvalue by ((k−2)√(k+1)+2)/(2k(k−1))·n − 1, showing the bound is tight for k ∈ {2,3,4,8,24}. The proof reduces the graph problem to the absolute projection constant of Banach space theory, settling the long-open case c₃ = 1/3 and Nikiforov's conjecture on sums of eigenvalues.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tightness for k=8,24 rests on unverified external equiangular-frame existence; upper-bound chain is internally sound.","rationale":"The reader identified the same weakest assumption: the tightness half depends on external facts about equiangular tight frames and the regular two-graph spectrum. My review confirms that the upper-bound chain (Theorems 2.1–2.3, 4.1, Lemmas 4.2–4.4, B.1) is internally consistent and the algebra checks out. The only load-bearing concern is the unproved existence of the maximal real equiangular tight frames in R^7 and R^23. Since these are known classical objects, the concern is addressable rather than a demonstrated error. The spectral identity is not a real concern because it follows directly from the frame properties. Therefore the reader's CONDITIONAL verdict remains appropriate; no verdict change is needed.","tokens_in":22323,"tokens_out":27754,"duration_ms":226885,"concrete_test":"Construct the 28-line equiangular tight frame in R^7 and the 276-line frame in R^23 explicitly from the E8 and Leech lattices. Verify for each frame that |<u_i,u_j>| = 1/√(r+2) for all i≠j and that U U^T = (N/r) I_r. Then form the sign matrix B and the 2N-vertex graph G0; diagonalize A(G0) and check that λ_{r+1}(G0) equals β_r/(2r)·2N − 1, where β_r = (r+√(r+2))/(1+√(r+2)). If both spectral checks pass, the tightness for k = 8 and k = 24 is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes equality c_k = α_k for k = 8 and k = 24. The proof of this equality in Section 5.1 imports two external facts: (i) existence of maximal real equiangular tight frames with N = (r+1)r/2 lines in R^r for r = 7 and r = 23, cited to [Gil18]; (ii) the spectral identity λ_{r+1}(G0) = β_r/(2r)|V(G0)|−1 for the associated regular two-graph, cited to [Sei76, GH92]. The second fact is actually a short derivation from the frame's Gram matrix and the block structure A(G0)+I = (1/2)[[J+B,J−B],[J−B,J+B]]; it is not a genuine risk. The first fact is the real dependency: without those frames, the lower-bound half collapses, leaving only the upper bound c_k ≤ α_k. The frames are classical (E8/Leech constructions), so the concern is about completeness of the proof rather than correctness. However, if the citation or construction were defective, the equalities c_8 = 5/28 and c_24 = 7/69 would not be established by this paper. The paper also cites Linz [Lin23] as 'in agreement,' but the equality claim is presented as independently derived from the frames.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22581,"tokens_out":20193,"duration_ms":173435,"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":[{"comment":"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.","section":"Section 5.1, paragraph 'The extremal equiangular line systems...'"},{"comment":"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.","section":"Section 5.1, 'The spectrum of this graph is well understood'"}],"minor_comments":[{"comment":"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.","section":"Proof of Theorem 2.1"},{"comment":"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.","section":"Lemma 4.3"},{"comment":"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.","section":"Table 1 and Section 5.3"},{"comment":"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.","section":"Appendix C"}],"recommendation":"major_revision","confidential_remarks":"The upper-bound half of the paper is sound and significant. My only substantive concern is the completeness of the tightness proof for k=8,24, which rests on unverified external equiangular-frame existence and an imported spectral identity. These are fixable within the manuscript's scope—by adding explicit constructions/references and the short derivation of the two-graph spectrum. If the authors do that, I would support acceptance. The numerical lower-bound table is computational evidence rather than a theorem; it should be presented as such."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this one if you care about Nikiforov's extremal eigenvalue program. The main theorem is real: λ_k(G) ≤ γ(k−1)n/(2(k−1)) − 1 ≤ α_k n − 1, which for k=3 gives c3=1/3 and closes the last open small-k case. The bound's 1/√k decay is the first of its kind, and Proposition 6.1 resolves Nikiforov's Conjecture 4.2 for sums of top eigenvalues.\n\nThe reduction in Theorem 2.2 is clean and appears to be new: Ky Fan for the trailing eigenvalues, then a trace estimate that lands exactly on the absolute projection constant. I re-derived the chain (trace estimate, Weyl step, Gram kernels, slack factorization) and it is internally consistent. The in-paper proof of the DL23 bound via Gegenbauer polynomials is a real alternative—the factorization in Lemma 4.4 is exact, and the slack identity in Lemma B.1 supports the strict improvement in Theorem 4.6, even if the quantitative gain is tiny.\n\nWhere the soft spots are: equality c_8=5/28 and c_24=7/69 imports existence of maximal real equiangular tight frames in R^7 and R^23 (E8/Leech, cited to Gil18). The paper does not prove those frames. If you treat the theorem as fully self-contained, that is the one external load-bearing input. The two-graph spectral identity cited to Seidel and Godsil–Hensel is less of a concern—it is a short derivation from the frame Gram matrix, and the paper sketches it. The weighted-core table relies on heuristic searches and ancillary files; it is honest lower-bound evidence, not part of the main theorem. The paper says so itself. Also, the lower-bound half for k=4,8,24 is presented as in agreement with Linz but independently derived from the frames; if the frames are accepted, fine.\n\nThe citation pattern is appropriate and there is no circularity. They do not hide dependencies; the limitations section is candid.\n\nBottom line: this deserves serious refereeing. The upper bound and the k=3 resolution are enough on their own. The referee should check Appendix B's stability constants and ask the authors to state the frame-existence dependence explicitly in the main theorem. That is a conditional accept in spirit, not a desk reject.","headline":"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.","tokens_in":23339,"tokens_out":2517,"would_cite":true,"duration_ms":24524,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","15A18","46B20","52C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["graph eigenvalues","adjacency matrix","absolute projection constant","orthogonal projections","equiangular lines","regular two-graphs","extremal eigenvalue bounds","positive definite kernels"],"falsifier":"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.","tokens_in":22035,"feed_emoji":"📊","tokens_out":7063,"duration_ms":61825,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every graph's kth eigenvalue obeys a universal bound, tight for five k","feed_subtitle":"It settles the last open small-k case and ties the five extremal constants to equiangular-line geometry.","key_machinery":"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","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Universal graph eigenvalue bound, tight for five k's","Projection constants unlock sharp graph eigenvalue inequalities","Five k's where graph eigenvalue bound is exact","Adjacency eigenvalues obey projection-constant ceiling","New bound for graph eigenvalues, tight for k=2,3,4,8,24"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Universal graph eigenvalue bound, tight for five k's","Projection constants unlock sharp graph eigenvalue inequalities","Five k's where graph eigenvalue bound is exact","Adjacency eigenvalues obey projection-constant ceiling","New bound for graph eigenvalues, tight for k=2,3,4,8,24"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000562,"raw_usage":{"total_tokens":2527,"prompt_tokens":789,"completion_tokens":1738,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":1658}},"tokens_in":533,"tokens_out":1738,"duration_ms":12736,"temperature":1.0,"reasoning_tokens":1658,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:33:00.072345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}