{
  "abstract": "The \"$S_{n,n}$ conjecture\" asserts that there does not exist a simple graph on $n$ vertices with Laplacian spectrum $\\{0,1,2,\\ldots,n-1\\}$ for any integer $n \\geq 2$. This conjecture has already been proved when $2 \\leq n \\leq 15$ and when $n \\geq 6,649,688,933$. We prove all remaining cases and thus establish that the conjecture is true.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26895",
  "author_total": 1,
  "authors": [
    {
      "name": "Nathaniel Johnston"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b098x2c7zh8v2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26895",
  "fulltext_url": null,
  "id": "b098x2c7zh8v2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26895",
  "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": "W7214230211",
  "pdf_url": "https://arxiv.org/pdf/2609.26895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b098x2c7zh8v2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Laplacian $S_{n,n}$ conjecture is true",
  "title_text": "The Laplacian S_n,n conjecture is true",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/b098x2c7zh8v2",
  "venue": "arXiv",
  "versions": []
}