{
  "abstract": "Let $G$ be a simple graph of order $n\\ge2$. Let $e(G)$ and $ω(G)$ denote its number of edges and clique number, respectively. Let $λ_1(G)\\ge\\cdots\\geλ_n(G)$ be its adjacency eigenvalues. In 2007, Bollobás and Nikiforov conjectured that every graph $G\\neq K_n$ satisfies $λ_1^2(G)+λ_2^2(G)\\le2\\left(1-\\frac1{ω(G)}\\right)e(G).$ This conjecture was recently completely resolved by Coutinho, Liu, Spier, Tang and Zhang. For every edge $e\\in E(G)$, let $c_G(e)$ denote the maximum order of a clique containing $e$. Kannan, Kumar and Pragada proposed the edge-localized conjecture $λ_1^2(G)+λ_2^2(G)\\le2\\sum_{e\\in E(G)}\\left(1-\\frac1{c_G(e)}\\right)$ for every graph $G\\neq K_n$. This conjecture was subsequently proved by Shaik. In this paper, we provide complete structural characterizations of the equality cases in both the Bollobás-Nikiforov inequality and its edge-localized version.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36912",
  "author_total": 2,
  "authors": [
    {
      "name": "Chuling Zhang"
    },
    {
      "name": "Xiying Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/arzrxf7006x7w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36912",
  "fulltext_url": null,
  "id": "arzrxf7006x7w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36912",
  "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": "W7224129064",
  "pdf_url": "https://arxiv.org/pdf/2610.36912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/arzrxf7006x7w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A characterization of equality in the edge-localized Bollobás-Nikiforov inequality",
  "title_text": "A characterization of equality in the edge-localized Bollobás-Nikiforov inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/arzrxf7006x7w",
  "venue": "arXiv",
  "versions": []
}