{
  "id": "2101.07952",
  "version": "v1",
  "published": "2021-01-20T03:54:09.000Z",
  "updated": "2021-01-20T03:54:09.000Z",
  "title": "A best bound for $λ_2(G)$ to guarantee $κ(G) \\geq 2$",
  "authors": [
    "Wenqian Zhang",
    "Jianfeng Wang"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a connected $d$-regular graph with a given order and the second largest eigenvalue $\\lambda_2(G)$. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for $\\lambda_2(G)$ which guarantees that $\\kappa(G) \\geq t+1$, where $1 \\leq t \\leq d-1$ and $\\kappa(G)$ is the vertex-connectivity of $G$, which was also mentioned by Cioab\\u{a}. As a starting point, we solve this problem in the case $t =1$, and characterize all families of extremal graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-20T03:54:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "best bound",
      "best upper bound",
      "second largest eigenvalue",
      "extremal graphs",
      "private communication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}