{
  "id": "1609.00373",
  "version": "v1",
  "published": "2016-09-01T10:11:44.000Z",
  "updated": "2016-09-01T10:11:44.000Z",
  "title": "A study of the Structural Properties of finite $G$-graphs and their Characterisation",
  "authors": [
    "Lord Clifford Kavi"
  ],
  "comment": "This is an MPhil thesis presented to the University of Ghana",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $G$-graph $\\Gamma(G,S)$ is a graph from the group $G$ generated by $S\\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\\langle s \\rangle, s\\in S$ with $k$-edges between two distinct cosets if there is an intersection of $k$ elements. In this thesis, after presenting some important properties of $G$-graphs, we show how the $G$-graph depends on the generating set of the group. We give the $G$-graphs of the symmetric group, alternating group and the semi-dihedral group with respect to various generating sets. We give a characterisation of finite $G$-graphs; in the general case and a bipartite case. Using these characterisations, we give several classes of graphs that are $G$-graphs. For instance, we consider the Tur\\'{a}n graphs, the platonic graphs and biregular graphs such as the Levi graphs of geometric configurations. We emphasis the structural properties of $G$-graphs and their relations to the group $G$ and the generating set $S$. As preliminary results for further studies, we give the adjacency matrix and spectrum of various finite $G$-graphs. As an application, we compute the energy of these graphs. We also present some preliminary results on infinite $G$-graphs where we consider the $G$-graphs of the infinite group $SL_2(\\mathbb{Z})$ and an infinite non-Abelian matrix group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-01T10:11:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C50"
    ],
    "keywords": [
      "structural properties",
      "characterisation",
      "generating set",
      "preliminary results",
      "infinite non-abelian matrix group"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}