{
  "id": "0909.3910",
  "version": "v1",
  "published": "2009-09-22T05:35:50.000Z",
  "updated": "2009-09-22T05:35:50.000Z",
  "title": "Note on the energy of regular graphs",
  "authors": [
    "Xueliang Li",
    "Yiyang Li",
    "Yongtang Shi"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a simple graph $G$, the energy $\\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known inequality is that $\\mathcal{E}(G)\\leq \\lambda_1+\\sqrt{(n-1)(2m-\\lambda_1)}$, where $\\lambda_1$ is the spectral radius. If $G$ is $k$-regular, we have $\\mathcal{E}(G)\\leq k+\\sqrt{k(n-1)(n-k)}$. Denote $\\mathcal{E}_0=k+\\sqrt{k(n-1)(n-k)}$. Balakrishnan [{\\it Linear Algebra Appl.} {\\bf 387} (2004) 287--295] proved that for each $\\epsilon>0$, there exist infinitely many $n$ for each of which there exists a $k$-regular graph $G$ of order $n$ with $k< n-1$ and $\\frac{\\mathcal{E}(G)}{\\mathcal{E}_0}<\\epsilon$, and proposed an open problem that, given a positive integer $n\\geq 3$, and $\\epsilon>0$, does there exist a $k$-regular graph $G$ of order $n$ such that $\\frac{\\mathcal{E}(G)}{\\mathcal{E}_0}>1-\\epsilon$. In this paper, we show that for each $\\epsilon>0$, there exist infinitely many such $n$ that $\\frac{\\mathcal{E}(G)}{\\mathcal{E}_0}>1-\\epsilon$. Moreover, we construct another class of simpler graphs which also supports the first assertion that $\\frac{\\mathcal{E}(G)}{\\mathcal{E}_0}<\\epsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-22T05:35:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C90",
      "15A18",
      "92E10"
    ],
    "keywords": [
      "regular graph",
      "linear algebra appl",
      "spectral radius",
      "absolute values",
      "adjacency matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.3910L"
    }
  }
}