{
  "abstract": "The Bollobás-Nikiforov (BN) conjecture, in its clique-number form, asserts that every non-complete graph $G$ of order $n$ with $m$ edges and clique number $\\omega(G)$ satisfies\\[\\lambda_1(G)^2+\\lambda_2(G)^2\\le 2\\left(1-\\frac1{\\omega(G)}\\right)m,\\]where $\\lambda_1(G)\\ge \\lambda_2(G)\\ge\\cdots\\ge\\lambda_n(G)$ are the adjacency eigenvalues of $G$.We prove an unconditional perturbative form for graphs that can be made triangle-free by deleting few edges. Let $\\eta_{\\triangle}(G)$ be the minimum number of edges whose deletion destroys all triangles of $G$. Every non-complete graph $G$ on at least two vertices satisfies\\[\\lambda_1(G)^2+\\lambda_2(G)^2\\le 2\\left(1-\\frac1\\omega\\right)m+C_\\omega \\eta_{\\triangle}(G),\\]where $\\omega=\\omega(G)$ and\\[C_\\omega=2+\\frac2\\omega+\\frac{4\\omega}{\\omega-2}\\qquad (\\omega\\ge3).\\]For $\\omega=1,2$, no additive error is needed. Since $C_\\omega\\le 44/3$, a uniform error term $(44/3) \\eta_{\\triangle}(G)$ is valid. The emphasis of the result is the explicit finite error that is linear in the edge-deletion distance from the triangle-free class. We also show that the vertex-disjoint triangle packing number cannot replace $\\eta_{\\triangle}(G)$ in this perturbative framework, and we clarify the relation with bounds expressed in terms of the total number of triangles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Yong Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/14qc0cxmf7nby.bib",
  "cited_by": 0,
  "doi": "10.37236/15969",
  "fulltext_url": null,
  "id": "14qc0cxmf7nby",
  "issns": [
    "1077-8926"
  ],
  "landing_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p27",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i4p27/pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-06",
  "publisher": "The Electronic Journal of Combinatorics",
  "ris": "https://research.draconitic.com/paper/14qc0cxmf7nby.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T12:20:56Z",
      "native_id": "10.37236/15969",
      "source": "crossref",
      "url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p27"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:1077-8926",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Robust Bollobás-Nikiforov Bound for Graphs Close to Triangle-Free Graphs",
  "title_text": "A Robust Bollobás-Nikiforov Bound for Graphs Close to Triangle-Free Graphs",
  "type": "peer-reviewed",
  "updated_at": "2026-11-08T00:21:33Z",
  "url": "https://research.draconitic.com/paper/14qc0cxmf7nby",
  "venue": "The Electronic Journal of Combinatorics",
  "versions": []
}