{ "schemaVersion": "1.6", "slug": "nordhaus-gaddum-laplacian-products", "title": "Nordhaus–Gaddum products of Laplacian eigenvalues: a counterexample and the sharp limiting profile", "shortTitle": "A graph and its complement: the Laplacian product bound fails", "url": "https://evidencepress.org/releases/nordhaus-gaddum-laplacian-products/", "oneLine": "A 2025 conjecture bounded the product of the k-th Laplacian eigenvalue of a graph and of its complement by n(n−k); a 14-vertex graph built from the Fano plane gives 115.85 > 112, and the true largest product is found to leading order, with a sudden drop at the middle rank.", "abstract": "Chen, Guo, Li and Wang (2025) conjectured that every graph G on n vertices satisfies μₖ(G)μₖ(Ḡ) ≤ n(n−k) for 1 ≤ k ≤ 3n/4, where μₖ is the k-th largest Laplacian eigenvalue. This paper disproves the inequality. An explicit 14-vertex graph built from the Fano plane gives μ₆(G)μ₆(Ḡ) = 63 + 7√57 > 112. Symmetric designs give an exact product formula and infinite families, and a design-free construction gives counterexamples whenever 2(k+1) ≤ n < ⌊(k+1)²/4⌋. Writing M(n,k) for the largest possible product, the limit of M(n,k)/n² is determined for every rank proportion θ = k/n ≠ 1/2: it is (1−θ/2)² below one half and 1/4 above. At θ = 1/2 the limit jumps from 9/16 to 1/4, and along the odd central sequence n = 2k+1 every limit point lies in [1/2, 9/16]. Necessary structure for asymptotic extremisers is also given. A floating-point enumeration of all graphs through order 10 found no violation exceeding 10⁻⁹. The inequality stays undecided at some pairs (n,k) with k < n/2. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-04", "dateModified": "2026-11-04", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23942304", "doiUrl": "https://doi.org/10.5281/zenodo.23942304", "conceptDoi": "10.5281/zenodo.23942303", "pdfUrl": "https://github.com/ipitchford/nordhaus-gaddum-laplacian-products/releases/download/v0.1.0-candidate/nordhaus-gaddum-laplacian-products-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23942304", "repoUrl": "https://github.com/ipitchford/nordhaus-gaddum-laplacian-products", "releaseUrl": "https://github.com/ipitchford/nordhaus-gaddum-laplacian-products/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/nordhaus-gaddum-laplacian-products/index.md", "bibtexUrl": "https://evidencepress.org/releases/nordhaus-gaddum-laplacian-products/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/nordhaus-gaddum-laplacian-products.mp3", "imageUrl": "https://evidencepress.org/assets/og/nordhaus-gaddum-laplacian-products.png?v=70a2b397af", "coverArtUrl": "https://evidencepress.org/assets/art/nordhaus-gaddum-laplacian-products.svg?v=3ced11a2c6", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/nordhaus-gaddum-laplacian-products.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/nordhaus-gaddum-laplacian-products.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematics candidate. The counterexample and limiting profile have written proofs; the n ≤ 10 enumeration is numerical and Conjecture 7.2 is open. Internal AI-assisted review with a second-model adversarial review, one revision round with extensions, proof audits and a re-review. No human peer review, independent reproduction or formal verification has been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23942304", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "code/run_checks.py was rerun from a fresh extraction of the published archive with Python -I -B and with -O (CPython 3.11.14, NumPy 2.4.6, SciPy 1.17.1, SymPy 1.14.0); both passed and outputs matched the shipped results. GitHub CI on Linux reruns both modes." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Internal reviewers recomputed values, but this is AI-assisted checking, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Package versions are pinned in requirements.txt; not a locked container. The shipped enumeration needs nauty geng 2.9.3 to repeat. No external data are used." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematics and computer verification", "AI-assisted internal review, second-model adversarial review, proof audits and re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript of 4 November 2026 (13 pages), its README, prior-art note, agent index and recorded outputs, and the replay of the check suite from a fresh extraction of the published archive.", "grandfatheredAtSchemaVersion": null, "keywords": [ "spectral graph theory", "Laplacian eigenvalues", "Nordhaus–Gaddum inequalities", "graph complement", "counterexample", "symmetric designs", "Fano plane", "extremal graph theory" ], "keyResults": [ "Counterexample (Section 2, proof): a 14-vertex graph built from the Fano plane has μ₆(G)μ₆(Ḡ) = 63 + 7√57 ≈ 115.85 > 112, disproving Conjecture 20 of Chen, Guo, Li and Wang.", "Families (Sections 3–5, proof): an exact product formula for symmetric designs, infinite design families, and design-free failures whenever 2(k+1) ≤ n < ⌊(k+1)²/4⌋.", "Limiting profile (Sections 6–7, proof): M(n,k)/n² tends to (1−θ/2)² below θ = 1/2 and 1/4 above, jumping from 9/16 to 1/4 at the centre.", "Odd central rank (Theorem 7.1, proof): at n = 2k+1 every limit point lies in [1/2, 9/16]; the limit 1/2 is conjectured.", "Structure (Section 8, proof): necessary conditions on asymptotic extremisers." ], "reviews": [], "evidencePackage": "Written proofs, not machine-verified. code/run_checks.py checks the 14-vertex graph with exact integer characteristic polynomials against shipped certificates, makes exact sign checks of the failure range for three binary designs and exact spectrum checks of odd-central joins for k ≤ 8, runs 460 finite numerical checks of the bounds, validates the shipped enumeration of all 12,293,433 graphs with n ≤ 10, and repeats that enumeration through n = 8 when geng is present. The suite was rerun from a fresh extraction of the published archive with Python -I -B and with -O; both passed and matched the shipped results. Corrupted copies fail under -O. GitHub CI on Linux reruns both modes.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Chen, Guo, Li and Wang (2025), A Conjecture on the Nordhaus-Gaddum Product Type Inequality for Laplacian Eigenvalue of a Graph. The Electronic Journal of Combinatorics. Source of Conjecture 20, disproved here; proves k = 1 and n/2 ≤ k ≤ 3n/4.", "url": "https://doi.org/10.37236/14060", "doi": "10.37236/14060" }, { "citation": "Li and Guo (2024), Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues. The Electronic Journal of Combinatorics. The earlier sum version, not resolved here.", "url": "https://doi.org/10.37236/12008", "doi": "10.37236/12008" }, { "citation": "Gibson (2026), An endpoint exception to a conjectured equality characterization for Laplacian eigenvalue products. arXiv. Refutes the equality clause and leaves k < n/2 open.", "url": "https://arxiv.org/abs/2609.16049", "doi": "10.48550/arXiv.2609.16049" }, { "citation": "van Lint and Seidel (1966), Equilateral point sets in elliptic geometry. Indagationes Mathematicae (Proceedings). Classical source of the Seidel matrix used in the upper bound.", "url": "https://doi.org/10.1016/s1385-7258(66)50038-5", "doi": "10.1016/s1385-7258(66)50038-5" }, { "citation": "Anonymous (2026), The homogeneous adjacency spectrum of the Fano plane. Evidence Press. See also: same plane, different object (hypergraph adjacency tensor).", "url": "https://doi.org/10.5281/zenodo.22850084", "doi": "10.5281/zenodo.22850084" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:nordhaus-gaddum-laplacian-products", "attemptIds": [ "ep-attempt:nordhaus-gaddum-laplacian-products-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "decision-rule", "description": "Whether the bound n(n−k) holds for the product of k-th Laplacian eigenvalues of a graph and its complement, and what replaces it: failure at the pairs of (2), (3) and (6)–(7), and the limiting profile (4).", "scope": "Simple graphs, combinatorial Laplacian. Some pairs (n,k) with k < n/2 stay undecided; finite-n bounds are not sharp." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Section 1 states the conjecture in the conventions of Chen, Guo, Li and Wang; Section 10 maps each prior unit to its source.", "remainingRisks": [ "The literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a spectral graph theorist check Sections 6–8 and ask the conjecture’s authors whether the inequality has been refuted elsewhere.", "claimCeiling": "Written proofs and an exact 14-vertex certificate; floating-point enumeration through n = 10; finite computer checks. 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