{
  "abstract": "Haemers' conjecture for the generalized spectrum asserts that almost all graphs are determined by their generalized spectrum, that is, for almost all graphs $G$, every graph with the same spectrum as $G$ whose complement has the same spectrum as the complement of $G$ is isomorphic to $G$. We prove this conjecture by showing that the random graph $G(n,1/2)$ is determined by its generalized spectrum with probability tending to one. The proof combines a local theory of the denominators of the rational orthogonal matrices that relate generalized cospectral graphs, Fourier analysis of random symmetric matrices over finite rings, Seidel switching, inverse Littlewood--Offord theory over finite fields, rank estimates for random symmetric matrices modulo primes, and a study of the rational factors of the characteristic polynomial of a random symmetric $\\pm1$ matrix with zero diagonal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11047",
  "author_total": 1,
  "authors": [
    {
      "name": "Ziqing Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/acgzqb9jjaz2d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11047",
  "fulltext_url": null,
  "id": "acgzqb9jjaz2d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11047",
  "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": "W7221122351",
  "pdf_url": "https://arxiv.org/pdf/2610.11047",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/acgzqb9jjaz2d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11047",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11047"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Almost all graphs are determined by their generalized spectrum",
  "title_text": "Almost all graphs are determined by their generalized spectrum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/acgzqb9jjaz2d",
  "venue": "arXiv",
  "versions": []
}