{
  "id": "1612.05043",
  "version": "v1",
  "published": "2016-12-15T12:34:21.000Z",
  "updated": "2016-12-15T12:34:21.000Z",
  "title": "Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph",
  "authors": [
    "Yong Lu",
    "Ligong Wang",
    "Qiannan Zhou"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G^{\\sigma}$ be an oriented graph and $S(G^{\\sigma})$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^{\\sigma}$. The skew-rank of $G^{\\sigma}$, denoted by $sr(G^{\\sigma})$, is the rank of $S(G^{\\sigma})$. Denote by $d(G)=|E(G)|-|V(G)|+\\theta(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\\theta(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^{\\sigma})\\leq r(G)+2d(G)$ for an oriented graph $G^{\\sigma}$, where $r(G)$ is the rank of the adjacency matrix of $G$, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of $sr(G^{\\sigma})$ of an oriented graph $G^{\\sigma}$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^{\\sigma})\\geq r(G)-2d(G)$ for an oriented graph $G^{\\sigma}$ and characterize the graphs whose skew-rank attain the lower bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-15T12:34:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "oriented graph",
      "cycle space",
      "skew-rank attain",
      "lower bound",
      "left open till"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}