{
  "abstract": "For a tree T, let gamma(T) be its domination number and let mu(T) count the Laplacian eigenvalues in [0,1). The strict inequality gamma(T)/mu(T)<4/3 leaves open three linked questions: the exact finite-order defect, the structure near equality, and whether a fixed defect permits only finitely many local obstructions. We answer all three. For every n>=2, max_{|V(T)|=n} (7 gamma(T)-9 mu(T)) = floor((n-20)/9). Thus order twenty-nine is the first at which the ratio 9/7 can be exceeded. The proof is based on an exact decomposition of Phi(T)=|V(T)|-20-9(7 gamma(T)-9 mu(T)) into nonnegative integer terms, together with a clean-contraction identity. For subcubic trees, equality cases are canonical lifts of trees with perfect matchings. We classify every layer through Phi=42, locating successive phase transitions at 20, 21, 40, 41, and 42. The finiteness mechanism is simple to state: after the repeatable matched-skeleton pieces are removed, fixed slack leaves only a bounded exceptional core. Formally, a local compactness theorem bounds every nonordinary weighted deep component at fixed packing surplus and residual negative index. Consequently, for each fixed k, every subcubic tree with Phi(T)=k consists of a bounded ported defect kernel, an arbitrary compatible forest of ordinary tiles, and at most floor(k/21) clean expansions. All unbounded statements are proved symbolically; exact computation is used only for explicitly bounded atom lists and independent finite verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27602",
  "author_total": 1,
  "authors": [
    {
      "name": "Yufeng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cw9cr4cm7ceg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27602",
  "fulltext_url": null,
  "id": "1cw9cr4cm7ceg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27602",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7204824039",
  "pdf_url": "https://arxiv.org/pdf/2608.27602",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cw9cr4cm7ceg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27602",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27602"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp Order Laws and Finite-Defect Stability for Domination versus Small Laplacian Eigenvalues in Trees",
  "title_text": "Sharp Order Laws and Finite-Defect Stability for Domination versus Small Laplacian Eigenvalues in Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/1cw9cr4cm7ceg",
  "venue": "arXiv",
  "versions": []
}