{
  "abstract": "Let $m_T[0,1)$ denote the number of Laplacian eigenvalues of a tree $T$ that are strictly less than $1$. Guo, Xue and Liu \\cite{GXL} proved that every tree $T$ of diameter $d$ satisfies $m_T[0,1)\\ge \\lceil(d+1)/3\\rceil$, and that this bound is sharp when $d\\equiv2\\pmod3$. It is classical that every graph of diameter $d$ has domination number at least $\\lceil(d+1)/3\\rceil$. In this paper, for any diameter $d$, $m_T[0,1)=\\lceil(d+1)/3\\rceil$ if and only if $γ(T)=\\lceil(d+1)/3\\rceil$; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy $m_T[0,1)\\ge\\lceil(d+1)/3\\rceil+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06899",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiaxin Guo"
    },
    {
      "name": "Tao Hu"
    },
    {
      "name": "Quanyu Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8synw6xxhdzex.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06899",
  "fulltext_url": null,
  "id": "8synw6xxhdzex",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06899",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7220977962",
  "pdf_url": "https://arxiv.org/pdf/2610.06899",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8synw6xxhdzex.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.06899",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06899"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The number of Laplacian eigenvalues of trees less than one",
  "title_text": "The number of Laplacian eigenvalues of trees less than one",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/8synw6xxhdzex",
  "venue": "arXiv",
  "versions": []
}