{
  "abstract": "The uniform intersection number of a finite group $G$, denoted by $ui(G)$, is the smallest positive integer $k$ such that the intersection of any $k$ distinct maximal subgroups of $G$ is equal to its Frattini subgroup. We determine the uniform intersection number of dicyclic, dihedral, and quasi-dihedral groups. We then investigate the uniform intersection number through the subgroup-based generating graph $\\mathcal{G}(G)$ of a finite group $G$. We characterize the finite groups $G$ for which both the intersection number and the uniform intersection number are equal to two, and show that this condition is equivalent to $\\mathcal{G}(G)$ being complete multipartite. We also classify finite groups in terms of the diameter, girth, and perfectness of $\\mathcal{G}(G)$. Finally, we show that $\\mathcal{G}(\\mathbb{Z}_n)$ is Hamiltonian if and only if $n=p_1^{r}p_2^{r}$, where $r\\geq 2$ and $p_1$ and $p_2$ are distinct primes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22268",
  "author_total": 2,
  "authors": [
    {
      "name": "Projesh Nath Choudhury"
    },
    {
      "name": "Biswaranjan Khanra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd9w6dgzschcg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22268",
  "fulltext_url": null,
  "id": "bd9w6dgzschcg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22268",
  "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"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7222481848",
  "pdf_url": "https://arxiv.org/pdf/2610.22268",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd9w6dgzschcg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22268",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22268"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups",
  "title_text": "Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/bd9w6dgzschcg",
  "venue": "arXiv",
  "versions": []
}