{
  "abstract": "Determining the number of realisations, up to isometries, of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that, asymptotically almost surely, the $d$-dimensional complex realisation number for each $n$-vertex graph in an Erdős-Rényi random graph process is either infinite or equal to $2^{n-t}$ where $t$ is the size of the $(d+1)$-core; moreover this number coincides exactly with the real realisation number for such graphs. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ positive semi-definite matrix completion problem with randomly selected non-diagonal unknown entries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.18487",
  "author_total": 3,
  "authors": [
    {
      "name": "Sean Dewar"
    },
    {
      "name": "Anthony Nixon"
    },
    {
      "name": "Ben Smith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ptx9qwc2ynw5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.18487",
  "fulltext_url": null,
  "id": "5ptx9qwc2ynw5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.18487",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7161687726",
  "pdf_url": "https://arxiv.org/pdf/2605.18487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ptx9qwc2ynw5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2605.18487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.18487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The number of realisations of a random graph",
  "title_text": "The number of realisations of a random graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/5ptx9qwc2ynw5",
  "venue": "arXiv",
  "versions": []
}