{
  "id": "1605.03530",
  "version": "v1",
  "published": "2016-05-11T17:41:26.000Z",
  "updated": "2016-05-11T17:41:26.000Z",
  "title": "Symmetric spreads of complete graphs",
  "authors": [
    "Teng Fang",
    "Xin Gui Fang",
    "Binzhou Xia",
    "Sanming Zhou"
  ],
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "A graph $\\Gamma$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\\Gamma$ with $V(\\Gamma)$ admitting a nontrivial $G$-invariant partition $\\mathcal{B}$ such that there is exactly one edge of $\\Gamma$ between any two distinct blocks of $\\mathcal{B}$. This is achieved by giving a classification of $(G, 2)$-point-transitive and $G$-block-transitive designs $\\mathcal{D}$ together with $G$-orbits $\\Omega$ on the flag set of $\\mathcal{D}$ such that $G_{\\sigma, L}$ is transitive on $L \\setminus \\{\\sigma\\}$ and $L \\cap N = \\{\\sigma\\}$ for distinct $(\\sigma, L), (\\sigma, N) \\in \\Omega$, where $G_{\\sigma, L}$ is the setwise stabilizer of $L$ in the stabilizer $G_{\\sigma}$ of $\\sigma$ in $G$. Along the way we determine all imprimitive blocks of $G_{\\sigma}$ on $V \\setminus \\{\\sigma\\}$ for every $2$-transitive group $G$ on a set $V$, where $\\sigma \\in V$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-11T17:41:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complete graphs",
      "symmetric spreads",
      "adjacent vertices",
      "symmetric graphs",
      "stabilizer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}