{
  "id": "1609.00835",
  "version": "v1",
  "published": "2016-09-03T15:14:49.000Z",
  "updated": "2016-09-03T15:14:49.000Z",
  "title": "On the $A_α$-spectra of trees",
  "authors": [
    "Vladimir Nikiforov",
    "Germain Pastén",
    "Oscar Rojo",
    "Ricardo L. Soto"
  ],
  "comment": "19 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 $\\alpha\\in\\left[ 0,1\\right],$ define the matrix $A_{\\alpha}\\left(G\\right) $ as \\[ A_{\\alpha}\\left(G\\right) =\\alpha D\\left(G\\right) +(1-\\alpha)A\\left(G\\right) \\] where $0\\leq\\alpha\\leq1$. This paper gives several results about the $A_{\\alpha}$-matrices of trees. In particular, it is shown that if $T_{\\Delta}$ is a tree of maximal degree $\\Delta,$ then the spectral radius of $A_{\\alpha}(T_{\\Delta})$ satisfies the tight inequality \\[ \\rho(A_{\\alpha}(T_{\\Delta}))<\\alpha\\Delta+2(1-\\alpha)\\sqrt{\\Delta-1}. \\] This bound extends previous bounds of Godsil, Lov\\'asz, and Stevanovi\\'c. The proof is based on some new results about the $A_{\\alpha}$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_{\\alpha}$ of general graphs are proved, implying tight bounds for paths and Bethe trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-03T15:14:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A48"
    ],
    "keywords": [
      "spectral radius",
      "general graphs",
      "generalized bethe trees",
      "adjacency matrix",
      "tight inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}