{
  "id": "2311.04486",
  "version": "v1",
  "published": "2023-11-08T06:37:57.000Z",
  "updated": "2023-11-08T06:37:57.000Z",
  "title": "On the diameter of Engel graphs",
  "authors": [
    "Andrea Lucchini",
    "Pablo Spiga"
  ],
  "comment": "8 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Given a finite group $G$, the Engel graph of $G$ is a directed graph $\\Gamma(G)$ encoding pairs of elements satisfying some Engel word. Namely, $\\Gamma(G)$ is the directed graph, where the vertices are the non-hypercentral elements of $G$ and where there is an arc from $x$ to $y$ if and only if $[x,_ n y] = 1$ for some $n \\in \\mathbb{N}$. From previous work, it is known that, except for a few exceptions, $\\Gamma(G)$ is strongly connected. In this paper, we give an absolute upper bound on the diameter of $\\Gamma(G)$, when $\\Gamma(G)$ is strongly connected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-08T06:37:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "engel graph",
      "directed graph",
      "absolute upper bound",
      "finite group",
      "non-hypercentral elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}