{
  "id": "1404.7230",
  "version": "v1",
  "published": "2014-04-29T04:09:12.000Z",
  "updated": "2014-04-29T04:09:12.000Z",
  "title": "The skew-rank of oriented graphs",
  "authors": [
    "Xueliang Li",
    "Guihai Yu"
  ],
  "comment": "17 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "An oriented graph $G^\\sigma$ is a digraph without loops and multiple arcs, where $G$ is called the underlying graph of $G^\\sigma$. Let $S(G^\\sigma)$ denote the skew-adjacency matrix of $G^\\sigma$. The rank of the skew-adjacency matrix of $G^\\sigma$ is called the {\\it skew-rank} of $G^\\sigma$, denoted by $sr(G^\\sigma)$. The skew-adjacency matrix of an oriented graph is skew symmetric and the skew-rank is even. In this paper we consider the skew-rank of simple oriented graphs. Firstly we give some preliminary results about the skew-rank. Secondly we characterize the oriented graphs with skew-rank 2 and characterize the oriented graphs with pendant vertices which attain the skew-rank 4. As a consequence, we list the oriented unicyclic graphs, the oriented bicyclic graphs with pendant vertices which attain the skew-rank 4. Moreover, we determine the skew-rank of oriented unicyclic graphs of order $n$ with girth $k$ in terms of matching number. We investigate the minimum value of the skew-rank among oriented unicyclic graphs of order $n$ with girth $k$ and characterize oriented unicyclic graphs attaining the minimum value. In addition, we consider oriented unicyclic graphs whose skew-adjacency matrices are nonsingular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-29T04:09:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "05C50",
      "05C75"
    ],
    "keywords": [
      "oriented graph",
      "skew-adjacency matrix",
      "oriented unicyclic graphs attaining",
      "pendant vertices",
      "minimum value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.7230L"
    }
  }
}