{
  "id": "2309.17292",
  "version": "v1",
  "published": "2023-09-29T14:49:02.000Z",
  "updated": "2023-09-29T14:49:02.000Z",
  "title": "Spectral gap and embedded trees for the Laplacian of the Erdős-Rényi graph",
  "authors": [
    "Raphael Ducatez",
    "Renaud Rivier"
  ],
  "comment": "22 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "For the Erd\\H{o}s-R\\'enyi graph of size $N$ with mean degree $(1+o(1))\\frac{\\log N}{t+1}\\leq d\\leq(1-o(1))\\frac{\\log N}{t}$ where $t\\in\\mathbb{N}^{*}$, with high probability the smallest non zero eigenvalue of the Laplacian is equal to $2-2\\cos(\\pi(2t+1)^{-1})+o(1)$. This eigenvalue arises from a small subgraph isomorphic to a line of size $t$ linked to the giant connected component by only one edge.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-29T14:49:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "erdős-rényi graph",
      "spectral gap",
      "embedded trees",
      "smallest non zero eigenvalue",
      "small subgraph isomorphic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}