{
  "id": "1909.05609",
  "version": "v1",
  "published": "2019-09-12T12:52:27.000Z",
  "updated": "2019-09-12T12:52:27.000Z",
  "title": "On the spectral radius and the energy of eccentricity matrix of a graph",
  "authors": [
    "Iswar Mahato",
    "R. Gurusamy",
    "M. Rajesh Kannan",
    "S. Arockiaraj"
  ],
  "comment": "11 Pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The eccentricity matrix $\\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph, among the trees, in terms of invertibility of the associated eccentricity matrix. The largest eigenvalue of $\\varepsilon(G)$ is called the $\\varepsilon$-spectral radius, and the eccentricity energy (or the $\\varepsilon$-energy) of $G$ is the sum of the absolute values of the eigenvalues of $\\varepsilon(G)$. We establish some bounds for the $\\varepsilon$-spectral radius and characterize the extreme graphs. Two graphs are said to be $\\varepsilon$-equienergetic if they have the same $\\varepsilon$-energy. For any $n \\geq 5$, we construct a pair of $\\varepsilon$-equienergetic graphs on $n$ vertices, which are not $\\varepsilon$-cospectral.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-12T12:52:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral radius",
      "distance matrix",
      "equienergetic graphs",
      "star graph",
      "associated eccentricity matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}