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  "slug": "fkk-majority-partitions",
  "title": "Strict majority partitions and weighted certificates for normalized Laplacian gaps",
  "shortTitle": "Beyond majority counting",
  "url": "https://evidencepress.org/releases/fkk-majority-partitions/",
  "oneLine": "A sharp partition theorem, weighted certificates beyond every majority cut, and an exact test for adding a leaf.",
  "abstract": "A strict majority across a graph partition guarantees the sharp normalized-Laplacian threshold 2/(n−1), with equality fully classified. A graph-specific weighted certificate uses classical scaled diagonal dominance; an explicit infinite family proves that it reaches beyond all majority and minority partitions. The package also gives an exact pendant reduction and separates universal proofs from finite integer verification. The general Nordhaus–Gaddum conjecture remains unresolved.",
  "datePublished": "2026-10-13",
  "dateModified": "2026-10-13",
  "version": "0.1.0-candidate",
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    "Anonymous"
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  "problem": {
    "name": "Normalized-Laplacian Nordhaus–Gaddum conjecture",
    "url": "https://arxiv.org/abs/2304.01979"
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  "keywords": [
    "normalized Laplacian",
    "Nordhaus–Gaddum",
    "spectral graph theory",
    "majority partitions",
    "scaled diagonal dominance",
    "exact certificates"
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    "Strict-majority partitions imply λ₂ ≥ 2/(n−1), with equality exactly for one universal vertex joined to two equal cliques.",
    "A positive linear scaling witness supplies a quantitative weighted certificate; an infinite family lies outside every majority/minority partition but is certified.",
    "An exact Schur-complement criterion describes the effect of adding a pendant vertex."
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    {
      "citation": "Faught, Kempton and Knudson (2024), A Nordhaus–Gaddum type problem for the normalized Laplacian spectrum and graph Cheeger constant: the parent conjecture.",
      "url": "https://arxiv.org/abs/2304.01979",
      "doi": "10.48550/arXiv.2304.01979"
    },
    {
      "citation": "Kim and Madras (2025), A Nordhaus–Gaddum Problem for the Spectral Gap of a Graph: preceding structural bounds.",
      "url": "https://doi.org/10.1002/jgt.23253",
      "doi": "10.1002/jgt.23253"
    },
    {
      "citation": "Einollahzadeh and Karkhaneei (2021), On the lower bound of the sum of the algebraic connectivity of a graph and its complement: combinatorial-Laplacian bounds.",
      "url": "https://arxiv.org/abs/1901.02047",
      "doi": "10.48550/arXiv.1901.02047"
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