{
  "id": "1906.07948",
  "version": "v1",
  "published": "2019-06-19T07:25:31.000Z",
  "updated": "2019-06-19T07:25:31.000Z",
  "title": "Group-theoretic generalisations of vertex and edge connectivities",
  "authors": [
    "Yinan Li",
    "Youming Qiao"
  ],
  "comment": "14 pages; 3 figures",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $p$ be an odd prime. Let $P$ be a finite $p$-group of class $2$ and exponent $p$, whose commutator quotient $P/[P,P]$ is of order $p^n$. We define two parameters for $P$ related to central decompositions. The first parameter, $\\kappa(P)$, is the smallest integer $s$ for the existence of a subgroup $S$ of $P$ satisfying (1) $S\\cap [P,P]=[S,S]$, (2) $|S/[S,S]|=p^{n-s}$, and (3) $S$ admits a non-trivial central decomposition. The second parameter, $\\lambda(P)$, is the smallest integer $s$ for the existence of a central subgroup $N$ of order $p^s$, such that $P/N$ admits a non-trivial central decomposition. While defined in purely group-theoretic terms, these two parameters generalise respectively the vertex and edge connectivities of graphs: For a simple undirected graph $G$, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lov\\'asz (B. Braz. Math. Soc., 1989), there is a $p$-group of class $2$ and exponent $p$ $P_G$ that is naturally associated with $G$. Our main result shows that the vertex connectivity $\\kappa(G)$ is equal to $\\kappa(P_G)$, and the edge connectivity $\\lambda(G)$ is equal to $\\lambda(P_G)$. We also discuss the relation between $\\kappa(P)$ and $\\lambda(P)$ for a general $p$-group $P$ of class $2$ and exponent $p$, as well as the computational aspects of these parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-19T07:25:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "edge connectivity",
      "group-theoretic generalisations",
      "non-trivial central decomposition",
      "smallest integer",
      "first parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}