{
  "abstract": "The likelihood of a finite simple undirected graph $G$ on $n$ vertices is the probability that the uniform sequential attachment process, which at each step joins a new vertex to a uniformly random subset of uniformly random size of the vertices already present, outputs a graph isomorphic to $G$. Dervovic, Mocherla and Severini conjectured that the likelihood is minimised by the balanced complete bipartite graph. We prove that, among complete bipartite graphs of a given order, the balanced one uniquely minimises the likelihood. Exact computation shows that it also minimises over all graphs for every order from $6$ through $14$. At $n=15$, the exact global minimisers are $C_5[\\bar K_3]$ and its complement, giving the first counterexample. Completed exact searches show that the global minimisers at order $16$ are $K_{8,8}$ and its complement. At order $17$, the unique minimiser is a self-complementary mixed blow-up of $C_5$. At order $18$, the minimisers are $K_{9,9}$ and its complement. We show that the failure is not sporadic by proving that the likelihood of the balanced complete bipartite graph is $2^{-(1/2-1/(8\\ln 2)+o(1))n^2}$, whereas the minimum over all graphs of order $n$ is $2^{-(1/2+o(1))n^2}$, so the conjectured minimiser exceeds the minimum by a factor exponential in $n^2$. We also determine the Shannon entropy of the process to leading order, namely $n^2/(4\\ln 2)$ bits, which shows that the conjectured minimiser is in fact more likely than a typical output of the process. The proofs rest on a vertex deletion recurrence which evaluates the likelihood in time $O(2^n n)$ and which closes on the blow-ups of any fixed base graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19467",
  "author_total": 2,
  "authors": [
    {
      "name": "Simone Severini"
    },
    {
      "name": "Eric W. Weisstein"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4jq5fnd25ja5z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19467",
  "fulltext_url": null,
  "id": "4jq5fnd25ja5z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19467",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7203937957",
  "pdf_url": "https://arxiv.org/pdf/2608.19467",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4jq5fnd25ja5z.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2608.19467",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19467"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The minimum of the graph likelihood",
  "title_text": "The minimum of the graph likelihood",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/4jq5fnd25ja5z",
  "venue": "arXiv",
  "versions": []
}