{
  "id": "1611.01818",
  "version": "v1",
  "published": "2016-11-06T18:27:26.000Z",
  "updated": "2016-11-06T18:27:26.000Z",
  "title": "A note on the positive semidefinitness of $A_α(G)$",
  "authors": [
    "Vladimir Nikiforov",
    "Oscar Rojo"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\\alpha\\in\\left[ 0,1\\right] $, write $A_{\\alpha}\\left( G\\right) $ for the matrix \\[ A_{\\alpha}\\left( G\\right) =\\alpha D\\left( G\\right) +(1-\\alpha)A\\left( G\\right) . \\] Let $\\alpha_{0}\\left( G\\right) $ be the smallest $\\alpha$ for which $A_{\\alpha}(G)$ is positive semidefinite. It is known that $\\alpha_{0}\\left( G\\right) \\leq1/2$. The main results of this paper are: (1) if $G$ is $d$-regular then \\[ \\alpha_{0}=\\frac{-\\lambda_{\\min}(A(G))}{d-\\lambda_{\\min}(A(G))}, \\] where $\\lambda_{\\min}(A(G))$ is the smallest eigenvalue of $A(G)$; (2) $G$ contains a bipartite component if and only if $\\alpha_{0}\\left( G\\right) =1/2$; (3) if $G$ is $r$-colorable, then $\\alpha_{0}\\left( G\\right) \\geq1/r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-06T18:27:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A48"
    ],
    "keywords": [
      "positive semidefinitness",
      "smallest eigenvalue",
      "adjacency matrix",
      "main results",
      "diagonal matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}