{
  "id": "1803.00246",
  "version": "v1",
  "published": "2018-03-01T08:35:49.000Z",
  "updated": "2018-03-01T08:35:49.000Z",
  "title": "Cographs: Eigenvalues and Dilworth Number",
  "authors": [
    "Ebrahim Ghorbani"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph $G$ is called the Dilworth number of $G$. We prove that for any cograph $G$, the multiplicity of any eigenvalue $\\lambda\\ne0,-1$, does not exceed the Dilworth number of $G$ and show that this bound is best possible. We give a new characterization of the family of cographs based on the existence of a basis consisting of weight 2 vectors for the eigenspace of either $0$ or $-1$ eigenvalues. This partially answers a question of G. F. Royle (2003).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-01T08:35:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C75"
    ],
    "keywords": [
      "dilworth number",
      "eigenvalue",
      "vicinal preorder",
      "vertex set",
      "simple graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}