{
  "id": "1709.00182",
  "version": "v1",
  "published": "2017-09-01T07:27:00.000Z",
  "updated": "2017-09-01T07:27:00.000Z",
  "title": "On the eigenvalues of $A_α$-spectra of graphs",
  "authors": [
    "Huiqiu Lin",
    "Jie Xue",
    "Jinlong Shu"
  ],
  "comment": "12 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 any real $\\alpha\\in [0,1]$, Nikiforov \\cite{VN1} defined the matrix $A_{\\alpha}(G)$ as $$A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G).$$ In this paper, we give some results on the eigenvalues of $A_{\\alpha}(G)$ with $\\alpha>1/2$. In particular, we show that for each $e\\notin E(G)$, $\\lambda_i(A_{\\alpha}(G+e))\\geq\\lambda_i(A_{\\alpha}(G))$. By utilizing the result, we prove have $\\lambda_k(A_{\\alpha}(G))\\leq\\alpha n-1$ for $2\\leq k\\leq n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that $\\lambda_n(A_{\\alpha}({G}))\\geq 2\\alpha-1$ if $G$ contains no isolated vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-01T07:27:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eigenvalues",
      "adjacency matrix",
      "diagonal matrix",
      "extremal graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}