{
  "id": "0709.3140",
  "version": "v1",
  "published": "2007-09-20T06:43:03.000Z",
  "updated": "2007-09-20T06:43:03.000Z",
  "title": "Some Relations between Rank, Chromatic Number and Energy of Graphs",
  "authors": [
    "S. Akbari",
    "E. Ghorbani",
    "S. Zare"
  ],
  "comment": "Accepted for publication in Discrete Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)={\\rm rank}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\\geq 2\\max(\\chi(G), n-\\chi(\\bar{G}))$, where $n$ is the number of vertices of $G$, $\\bar{G}$ and $\\chi(G)$ are the complement and the chromatic number of $G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of ${\\rm rank}(G)$ are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-20T06:43:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C50",
      "15A03"
    ],
    "keywords": [
      "chromatic number",
      "absolute values",
      "adjacency matrix",
      "lower bounds",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.3140A"
    }
  }
}