{
  "id": "1304.5707",
  "version": "v6",
  "published": "2013-04-21T08:40:33.000Z",
  "updated": "2015-05-18T05:27:43.000Z",
  "title": "A survey on the skew energy of oriented graphs",
  "authors": [
    "Xueliang Li",
    "Huishu Lian"
  ],
  "comment": "This will appear as a chapter in Mathematical Chemistry Monograph No.17: Energies of Graphs -- Theory and Applications, edited by I. Gutman and X. Li",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. The energy of $G$ is defined as the sum of absolute values of all eigenvalues of $A(G)$, which was introduced by Gutman in 1970s. Since graph energy has important chemical applications, it causes great concern and has many generalizations. The skew energy and skew energy-like are the generalizations in oriented graphs. Let $G^\\sigma$ be an oriented graph of $G$ with skew adjacency matrix $S(G^\\sigma)$. The skew energy of $G^\\sigma$, denoted by $\\mathcal{E}_S(G^\\sigma)$, is defined as the sum of the norms of all eigenvalues of $S(G^\\sigma)$, which was introduced by Adiga, Balakrishnan and So in 2010. In this paper, we summarize main results on the skew energy of oriented graphs. Some open problems are proposed for further study. Besides, results on the skew energy-like: the skew Laplacian energy and skew Randi\\'{c} energy are also surveyed at the end.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2014-03-04T10:45:28.000Z",
      "abstract": "The skew energy of an oriented graph was introduced by Adiga, Balakrishnan and So in 2010, as one of various generalizations of the energy of an undirected graph. Let $G$ be a simple undirected graph, and $G^\\sigma$ be an oriented graph of $G$ with the orientation $\\sigma$ and skew-adjacency matrix $S(G^\\sigma)$. The skew energy of the oriented graph $G^\\sigma$, denoted by $\\mathcal{E}_S(G^\\sigma)$, is defined as the sum of the norms of all the eigenvalues of $S(G^\\sigma)$. In this paper, we summarize most of the known results on the skew energy of oriented graphs. Some open problems are also proposed for further study.",
      "comment": "37 pages. Welcome to provide more papers on this subject so that we can update the survey. arXiv admin note: text overlap with arXiv:1108.6229 by other authors",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2015-05-18T05:27:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B20",
      "05C20",
      "05C35",
      "05C50",
      "05C80",
      "05C90",
      "15A18"
    ],
    "keywords": [
      "oriented graph",
      "skew energy",
      "simple undirected graph",
      "skew-adjacency matrix",
      "open problems"
    ],
    "tags": [
      "monograph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.5707L"
    }
  }
}