{
  "id": "1906.05851",
  "version": "v1",
  "published": "2019-06-13T17:50:29.000Z",
  "updated": "2019-06-13T17:50:29.000Z",
  "title": "Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups",
  "authors": [
    "C. Dalfó",
    "M. A. Fiol"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs $P(n)$ and to a recent known family of mixed graphs $\\Gamma(d,n,r)$ (having edges with and without direction). As a byproduct, the existence of perfect codes in $P(n)$ allows us to give a lower bound for the multiplicity of its eigenvalue $-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-13T17:50:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "regular partition",
      "permutation groups",
      "eigenspaces",
      "mixed graphs",
      "eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}