{
  "abstract": "In 1993, Cao and Hong posed the problem of characterizing graphs whose second largest adjacency eigenvalue is greater than $1/3$ and less than $(\\sqrt5-1)/2$. Subsequent work extended the question to the non-strict bound. We give a complete structural characterization of connected graphs $G$ with $|V(G)|\\ge2$ satisfying $λ_2(G)\\le(\\sqrt5-1)/2$. Their complements are described by disjoint unions of graphs from an explicit family with least adjacency eigenvalue at least $-(1+\\sqrt5)/2$, possibly together with a cone or a thin or thick spider constructed from disjoint unions of members of the same family, subject to explicit numerical inequalities. This provides a positive characterization of the class, going beyond the classical result of Cvetković and Simić that the bound $λ_2\\le(\\sqrt5-1)/2$ admits only finitely many minimal forbidden induced subgraphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02005",
  "author_total": 4,
  "authors": [
    {
      "name": "Chaohui Chen"
    },
    {
      "name": "Hangxi Cha"
    },
    {
      "name": "Haiying Shan"
    },
    {
      "name": "Muhuo Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aqpmefz2b6haf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02005",
  "fulltext_url": null,
  "id": "aqpmefz2b6haf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02005",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02005",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aqpmefz2b6haf.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02005",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02005"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A complete characterization of graphs with second largest eigenvalue at most $(\\sqrt5-1)/2$",
  "title_text": "A complete characterization of graphs with second largest eigenvalue at most (√5−1)/2",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/aqpmefz2b6haf",
  "venue": "arXiv",
  "versions": []
}