{
  "id": "2208.05575",
  "version": "v1",
  "published": "2022-08-10T21:56:12.000Z",
  "updated": "2022-08-10T21:56:12.000Z",
  "title": "On the distribution of eigenvalues of increasing trees",
  "authors": [
    "Kenneth Dadedzi",
    "Stephan Wagner"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We prove that the multiplicity of a fixed eigenvalue $\\alpha$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $\\mu_{\\alpha} n$ and $\\sigma^2_{\\alpha} n$ respectively. It is also shown that $\\mu_{\\alpha}$ and $\\sigma^2_{\\alpha}$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants $\\mu_0$ and $\\sigma^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-10T21:56:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C05"
    ],
    "keywords": [
      "distribution",
      "totally real algebraic integer",
      "central limit theorem",
      "binary increasing trees",
      "variance asymptotically equal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}