{
  "id": "2402.12884",
  "version": "v2",
  "published": "2024-02-20T10:25:04.000Z",
  "updated": "2024-08-20T16:09:54.000Z",
  "title": "Lower bounds for the Randić index in terms of matching number",
  "authors": [
    "Saieed Akbari",
    "Sina Ghasemi Nezhad",
    "Reyhane Ghazizadeh",
    "John Haslegrave",
    "Elahe Tohidi"
  ],
  "comment": "11 pages, 2 figures. Added the observation that the assumptions in Corollary 6 are necessary, and other minor remarks",
  "categories": [
    "math.CO"
  ],
  "abstract": "We investigate how small the Randi\\'c index of a graph can be in terms of its matching number, and prove several results. We give best-possible linear bounds for graphs of small excess and for subcubic graphs; in the former case the size of excess we permit is qualitatively the best possible. We show that a linear bound holds for any sparse hereditary graph class (such as planar graphs). In general, however, we show that it can be much smaller than linear. We determine the asymptotic growth rate of the minimum Randi\\'c index for graphs with a near perfect matching, and conjecture that the same bounds hold for all graphs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-08-20T16:09:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C09",
      "05C70",
      "05C35"
    ],
    "keywords": [
      "matching number",
      "lower bounds",
      "sparse hereditary graph class",
      "linear bound holds",
      "asymptotic growth rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}