{
  "abstract": "For a degree sequence d on n vertices with M edges, let G(d) be the number of simple graphs and D(d) the number of loopless digraphs with in-degree = out-degree = d. We conjecture that G(d) >= sqrt(D(d)) exp(-C n log(2+M/n)) for every graphical d, a single inequality across all densities, and prove it up to an extra factor log n: G(d) >= sqrt(D(d)) exp(-C n log n log(2+M/n)) for every graphical d. The proof compares pairs of graphs with Eulerian digraphs by alternately colouring closed trails; the failure of injectivity is measured exactly, shared edges cancel against 2-cycles, and parity is handled by merging trails and by a parity argument on Eulerian circuits of odd components. No asymptotic enumeration and no bound on the maximum-entropy multipliers is used. When the maximum-entropy edge probabilities lie in [delta, 1-delta], the same method gives the conjecture itself, G(d) >= sqrt(D(d)) exp(-C n log n / delta). Combined with the exact identity G(d) = exp(g*(d)) pi(d), where pi(d) is the probability that the maximum-entropy beta-model has degrees exactly d, and a van der Waerden-type bound for D(d), this shows that uniformly random graphs with degree sequence d are close in the normalized cut metric to their beta-model whenever M >> n log n log(M/n), at every density, with no tameness and no cap on the degrees. We also record the conjecture in the sparse regime and, with constant 1/2+o(1), in the dense tame regime, from the asymptotic enumerations of McKay-Wormald and Barvinok-Hartigan.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02191",
  "author_total": 1,
  "authors": [
    {
      "name": "Behzad Mehrdad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0eemeds8yj4rq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02191",
  "fulltext_url": null,
  "id": "0eemeds8yj4rq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02191",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0eemeds8yj4rq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Graphs versus loopless digraphs with a given degree sequence, and convergence of uniformly random graphs at every density",
  "title_text": "Graphs versus loopless digraphs with a given degree sequence, and convergence of uniformly random graphs at every density",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/0eemeds8yj4rq",
  "venue": "arXiv",
  "versions": []
}