{
  "abstract": "The difference graph $D(G)$ of a finite group $G$ is obtained from the edge difference between its intersection power graph and power graph, after deleting isolated vertices. This graph has already been studied, with sufficient conditions for connectedness and a diameter bound $6$ for finite groups satisfying those conditions. We use generalized graph composition to reduce $D(G)$ to a graph $B(G)$ on the cyclic subgroups of $G$, so that connectedness and diameter are determined by the subgroup structure of $G$. We obtain a general criterion for the non-emptiness of $B(G)$ in terms of branching subgroups and b-normality, and characterize its connectedness for finite $p$-groups, non-cyclic finite abelian groups, and non-abelian groups with both trivial and non-trivial center. Combined with the previously established cyclic-group case, this gives a complete characterization of non-emptiness and connectedness of difference graphs for all finite groups. The successive structural cases lead naturally to the sharp diameter bounds $2,3,4,$ and $5$. For centerless non-abelian groups, connectedness is governed either by a unique branching subgroup or by an auxiliary graph $\\mathcal A(G)$; in the latter case \\[ \\operatorname{diam}\\mathcal A(G)-1 \\leq \\operatorname{diam}B(G) \\leq \\max\\{4,\\operatorname{diam}\\mathcal A(G)+1\\}, \\] and both bounds are sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00080",
  "author_total": 3,
  "authors": [
    {
      "name": "Shamik Ghosh"
    },
    {
      "name": "Sanchita Paul"
    },
    {
      "name": "M. K. Sen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ap01wh6xc1kw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00080",
  "fulltext_url": null,
  "id": "7ap01wh6xc1kw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00080",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7206189503",
  "pdf_url": "https://arxiv.org/pdf/2609.00080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ap01wh6xc1kw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.00080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups",
  "title_text": "Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7ap01wh6xc1kw",
  "venue": "arXiv",
  "versions": []
}