{
  "id": "2012.11372",
  "version": "v1",
  "published": "2020-12-18T06:27:28.000Z",
  "updated": "2020-12-18T06:27:28.000Z",
  "title": "Families of Circulant Graphs Without CI-Property and More Abelian Groups",
  "authors": [
    "V. Vilfred Kamalappan"
  ],
  "comment": "This article is an extension and generalization of the paper: V. Vilfred Kamalappan, \\emph{ New Families of Circulant Graphs Without Cayley Isomorphism Property with $r_i = 2$}, Int. J. Appl. Comput. Math., (2020) 6:90, 34 pages. https://doi.org/10.1007/s40819-020-00835-0. Published online: 28.07.2020 Springer",
  "categories": [
    "math.CO"
  ],
  "abstract": "A circulant graph $C_n(R)$ is said to have the Cayley Isomorphism (CI) property if whenever $C_n(S)$ is isomorphic to $C_n(R),$ there is some $a\\in \\mathbb{Z}_n^*$ for which $S = aR$. In this paper, we obtain many families of Type-2 isomorphic circulant graphs and new abelian groups. Type-2 isomorphism of circulant graphs is a new type of isomorphism, different from already known Adam's or Type-1 isomorphism and Type-2 isomorphic circulant graphs have the property that they are without $CI$-property. The main results are Theorems \\ref{d4} and \\ref{d7}. Using Theorem \\ref{d7} and Lemma \\ref{d9}, a list of new abelian groups, $(T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)),~\\circ)$ are given in the annexure for $n$ = 1 to 5, $p$ = 3,5,7, $x$ = 1 to $p-1$, $y$ = 0 to $np-1$ and $p,np^3-p\\in R^{np^3,x+yp}_i$ and for $p$ = 11, $n$ = 1 to 2, $x$ = 1 to $p-1$, $y$ = 0 to $np-1$ and $p,np^3-p\\in R^{np^3,x+yp}_i$. For more clarity, a list, without using Lemma \\ref{d9}, is given in \\cite{vw0A} for $n$ = 1 to 5, $p$ = 3,5,7,11, $x$ = 1 to $p-1$, $y$ = 0 to $np-1$ and $p,np^3-p\\in R^{np^3,x+yp}_i$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-18T06:27:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C60",
      "05C25",
      "05C75"
    ],
    "keywords": [
      "abelian groups",
      "isomorphic circulant graphs",
      "ci-property",
      "main results",
      "cayley isomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}