{
  "abstract": "This article is the $6^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \\cite{v2-1}-\\cite{v2-10}. In this part, we define $V_{n,m}(C_n(R))$ and Type-2 set $T2_{n,m}(C_n(R))$ of $C_n(R)$ and present their properties. We prove that $(V_{n,m}(C_n(R)), \\circ)$ is an Abelian group and $(T2_{n,m}(C_n(R)), \\circ)$ is a subgroup of $(V_{n,m}(C_n(R)), \\circ)$ where $T2_{n,m}(C_n(R))$ = $\\{C_n(R)\\}$ $\\cup$ $\\{C_n(S):$ $C_n(S)$ is Typ-2 isomorphic to $C_n(R)$ w.r.t. $m \\}$ and $(T2_{n,m}(C_n(R)), \\circ)$ is the Type-2 group of $C_n(R)$ w.r.t. $m$. We also present many examples of Type-1 and Type-2 groups where $T1_{n}(C_n(R))$ = $\\{C_n(xR): x\\inφ_{n}\\}$ is the Type-1 set of $C_n(R)$ and $(T1_{n}(C_n(R)), \\circ')$ is its Type-1 group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.12868",
  "author_total": 1,
  "authors": [
    {
      "name": "Vilfred Kamalappan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d04vbzq0wme15.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.12868",
  "fulltext_url": null,
  "id": "d04vbzq0wme15",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.12868",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7161164812",
  "pdf_url": "https://arxiv.org/pdf/2605.12868",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d04vbzq0wme15.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2605.12868",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.12868"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A study on Type-2 isomorphic circulant graphs. Part 6: Abelian groups $(T2_{n,m}(C_n(R)), \\circ)$ and $(V_{n,m}(C_n(R)), \\circ)$",
  "title_text": "A study on Type-2 isomorphic circulant graphs. Part 6: Abelian groups (T2_n,m(C_n(R)), ∘) and (V_n,m(C_n(R)), ∘)",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/d04vbzq0wme15",
  "venue": "arXiv",
  "versions": []
}