{
  "id": "2106.01261",
  "version": "v1",
  "published": "2021-06-02T16:13:17.000Z",
  "updated": "2021-06-02T16:13:17.000Z",
  "title": "Integral mixed circulant graph",
  "authors": [
    "Monu Kadyan",
    "Bikash Bhattacharjya"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A mixed graph is said to be \\textit{integral} if all the eigenvalues of its Hermitian adjacency matrix are integer. The \\textit{mixed circulant graph} $Circ(\\mathbb{Z}_n,\\mathcal{C})$ is a mixed graph on the vertex set $\\mathbb{Z}_n$ and edge set $\\{ (a,b): b-a\\in \\mathcal{C} \\}$, where $0\\not\\in \\mathcal{C}$. If $\\mathcal{C}$ is closed under inverse, then $Circ(\\mathbb{Z}_n,\\mathcal{C})$ is called a \\textit{circulant graph}. We express the eigenvalues of $Circ(\\mathbb{Z}_n,\\mathcal{C})$ in terms of primitive $n$-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of $Circ(\\mathbb{Z}_n,\\mathcal{C})$. For $n\\equiv 0 \\Mod 4$, we factorize the cyclotomic polynomial into two irreducible factors over $\\mathbb{Q}(i)$. Using this factorization, we characterize integral mixed circulant graphs in terms of its symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-02T16:13:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C25"
    ],
    "keywords": [
      "eigenvalues",
      "ramanujan type sum",
      "mixed graph",
      "characterize integral mixed circulant graphs",
      "integral oriented circulant graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}