{
  "id": "2105.03466",
  "version": "v1",
  "published": "2021-05-07T18:52:55.000Z",
  "updated": "2021-05-07T18:52:55.000Z",
  "title": "Perron value and moment of rooted trees",
  "authors": [
    "Lorenzo Ciardo"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Perron value $\\rho(T)$ of a rooted tree $T$ has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for $T$. A different, combinatorial weight notion for $T$ - the moment $\\mu(T)$ - emerges from the analysis of Kemeny's constant in the context of random walks on graphs. In the present work, we compare these two weight concepts showing that $\\mu(T)$ is \"almost\" an upper bound for $\\rho(T)$ and the ratio $\\mu(T)/\\rho(T)$ is unbounded but at most linear in the order of $T$. To achieve these primary goals, we introduce two new objects associated with $T$ - the Perron entropy and the neckbottle matrix - and we investigate how different operations on the set of rooted trees affect the Perron value and the moment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-07T18:52:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C76",
      "05C05",
      "15A18",
      "05C81"
    ],
    "keywords": [
      "perron value",
      "combinatorial weight notion",
      "random walks",
      "spectral nature",
      "algebraic connectivity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}