{
  "id": "1808.01391",
  "version": "v1",
  "published": "2018-08-03T23:31:58.000Z",
  "updated": "2018-08-03T23:31:58.000Z",
  "title": "Spectra of Cayley graphs",
  "authors": [
    "Wenbin Guo",
    "Daria V. Lytkina",
    "Victor D. Mazurov",
    "Danila O. Revin"
  ],
  "comment": "in Russian",
  "categories": [
    "math.GR",
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let $G$ be a group and $S\\subseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}=\\{s^{-1}\\mid s\\in S\\}$. Then {\\it the Cayley graph ${\\rm Cay}(G,S)$} is an undirected graph $\\Gamma$ with the vertex set $V(\\Gamma)=G$ and the edge set $E(\\Gamma)=\\{(g,gs)\\mid g\\in G, s\\in S\\}$. A graph $\\Gamma$ is said to be {\\it integral} if every eigenvalue of the adjacency matrix of $\\Gamma$ is integer. In the paper, we prove the following theorem: {\\it if a subset $S=S^{-1}$ of $G$ is normal and $s\\in S\\Rightarrow s^k\\in S$ for every $k\\in \\mathbb{Z}$ such that $(k,|s|)=1$, then ${\\rm Cay}(G,S)$ is integral.} In particular, {\\it if $S\\subseteq G$ is a normal set of involutions, then ${\\rm Cay}(G,S)$ is integral.} We also use the theorem to prove that {\\it if $G=A_n$ and $S=\\{(12i)^{\\pm1}\\mid i=3,\\dots,n\\}$, then ${\\rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in \"Kourovka Notebook\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-03T23:31:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20C05",
      "20C15"
    ],
    "keywords": [
      "cayley graph",
      "edge set",
      "vertex set",
      "adjacency matrix",
      "normal set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "ru",
      "license": "arXiv",
      "status": "editable"
    }
  }
}