{
  "id": "2412.00635",
  "version": "v1",
  "published": "2024-12-01T01:22:47.000Z",
  "updated": "2024-12-01T01:22:47.000Z",
  "title": "Critical threshold for regular graphs",
  "authors": [
    "Ishaan Bhadoo"
  ],
  "comment": "7 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \\geq \\frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \\frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-01T01:22:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular graphs",
      "critical threshold",
      "regular tree",
      "critical percolation threshold",
      "well-known"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}