{
  "abstract": "This article is the first part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \\cite{v2-1}-\\cite{v2-10}. Circulant graphs $C_n(R)$ and $C_n(S)$ are said to be \\emph{Adam's isomorphic} if there exist some $a\\in \\mathbb{Z}_n^*$ such that $S = a R$ under arithmetic reflexive modulo $n$ \\cite{ad67}. In 1970, Elspas and Turner \\cite{eltu} raised a question on the isomorphism of $C_{16}(1,3,7)$ and $C_{16}(2,3,5)$. The author \\cite{v96} further modify the definition to $m$ and $m^3$ are divisors of $\\gcd(n, r)$ and $n$, respectively, and $r\\in R$. Using the modified definition \\ref{d16}, we present our study on Type-2 isomorphism of circulant graphs $C_n(R)$ w.r.t. $m$ = 2. We reprove that $(i)$ $C_{16}(1,2,7)$ and $C_{16}(2,3,5)$ are Type-2 isomorphic w.r.t. $m$ = 2; $(ii)$ For $n \\geq 2$, $k \\geq 3$, $1 \\leq 2s-1 \\leq 2n-1$, $n \\neq 2s-1$, $R$ = $\\{2, 2s-1, 4n-(2s-1)\\}$ and $S$ = $\\{2, 2n-(2s-1), 2n+2s-1\\}$, $C_{8n}(R)$ and $C_{8n}(S)$ are Type-2 isomorphic w.r.t. $m$ = 2, $n,s\\in\\mathbb{N}$; and $(iii)$ For $n \\geq 2$, $1 \\leq 2s-1 < 2s'-1 \\leq [\\frac{n}{2}]$, $0 \\leq t \\leq [\\frac{n}{2}]$, $R$ = $\\{2,2s-1, 2s'-1\\}$ and $n,s,s'\\in \\mathbb{N}$, if $θ_{n,2,t}(C_n(R))$ and $C_n(R)$ are isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 for some $t$, then $n \\equiv 0~(mod ~ 8)$, $2s-1+2s'-1$ = $\\frac{n}{2}$, $2s-1 \\neq \\frac{n}{8}$, $t$ = $\\frac{n}{8}$ or $\\frac{3n}{8}$, $1 \\leq 2s-1 \\leq \\frac{n}{4}$ and $n \\geq 16$ where $θ_{n,m,t}$ is a transformation used to define Type-2 isomorphism of a circulant graph. At the end, we present a VB program POLY215.EXE which shows how Type-2 isomorphism w.r.t. $m$ = 2 of $C_{8n}(R)$ takes place for $R = \\{2, 2s-1, 4n-(2s-1)\\}$, $n \\geq 2$ and $n,s\\in {\\mathbb N}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.11441",
  "author_total": 1,
  "authors": [
    {
      "name": "Vilfred Kamalappan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7ctzr0npwmwy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.11441",
  "fulltext_url": null,
  "id": "a7ctzr0npwmwy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.11441",
  "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",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7160993895",
  "pdf_url": "https://arxiv.org/pdf/2605.11441",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7ctzr0npwmwy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2605.11441",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.11441"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A study on Type-2 isomorphic circulant graphs. Part 1: Type-2 isomorphic circulant graphs $C_n(R)$ w.r.t. $m$ = 2",
  "title_text": "A study on Type-2 isomorphic circulant graphs. Part 1: Type-2 isomorphic circulant graphs C_n(R) w.r.t. m = 2",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/a7ctzr0npwmwy",
  "venue": "arXiv",
  "versions": []
}