{
  "abstract": "Let $λ_2(G)$ and $κ(G)$ denote the second largest adjacency eigenvalue and the vertex-connectivity of a graph $G$. For a non-complete $r$-regular graph $G$, Fiedler's inequality $λ_2(G)\\ge r-κ(G)$ shows that $λ_2(G)<r-t$ forces $κ(G)\\ge t+1$. The best such bound is known for $t=1$, by Zhang and Wang, and for $t=2$ with $r\\ge4$, by Zhang. In this paper, for every $r\\ge3$ and every integer $t$ with $2\\le t\\le r-1$, we give an upper bound on $λ_2(G)$ that forces an $r$-regular graph $G$ to satisfy $κ(G)\\ge t+1$, and we determine exactly when this bound is best possible. For $\\frac{r+2}{2}\\le t\\le r-1$, Fiedler's bound $r-t$ is already best possible. For $2\\le t<\\frac{r+2}{2}$, it is replaced by one of two explicit algebraic numbers, and the sign of an explicit polynomial $W(r,t)$, which never vanishes, decides which one applies. Each of the two numbers is attained exactly when $(r,t)$ satisfies an explicit arithmetic condition, and every extremal graph then consists of two cliques and an independent cut of size $t$. In this range our bound also improves a bound of Zhang that holds for every such $t$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33064",
  "author_total": 2,
  "authors": [
    {
      "name": "Suil O"
    },
    {
      "name": "Lukai Sui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/049nnn897qzm0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33064",
  "fulltext_url": null,
  "id": "049nnn897qzm0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33064",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7223978658",
  "pdf_url": "https://arxiv.org/pdf/2610.33064",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/049nnn897qzm0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33064",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33064"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Second largest eigenvalue thresholds for the vertex-connectivity of regular graphs",
  "title_text": "Second largest eigenvalue thresholds for the vertex-connectivity of regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/049nnn897qzm0",
  "venue": "arXiv",
  "versions": []
}