{
  "abstract": "For a fixed integer $k\\ge 2$, let $kK_3$ denote the vertex-disjoint union of $k$ triangles. A recent fixed-size spectral theorem of Das and Yamini asserts that, for all sufficiently large $m$, every $kK_3$-free graph $G$ of size $m$ satisfies $λ(G)\\le (k-1)+\\sqrt{m-k(k-1)},$ and equality holds if and only if $(2k-1)\\mid m$ and $ G\\cong \\bigl(K_{2k-1}\\vee qK_1\\bigr)\\cup tK_1,\\, q=\\frac{m}{2k-1}-(k-1) $ for some $t\\ge 0$. They explicitly posed the open problem: Let $k\\ge 2$ be fixed and $\\ell$ be a residue in $\\{1,\\dots,2k-2\\}$. For all sufficiently large integers $m\\equiv \\ell\\pmod{2k-1}$, determine the exact value of $\\max\\bigl\\{λ(G): e(G)=m,\\ G\\text{ is }kK_3\\text{-free}\\bigr\\}, $ and characterize all graphs attaining this maximum. In this paper, using the positive-defect version of the bounded-core method for divisible sizes together with several new ideas developed in this paper, we give a complete solution to the aforementioned open problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07022",
  "author_total": 3,
  "authors": [
    {
      "name": "Jing Gao"
    },
    {
      "name": "Shuchao Li"
    },
    {
      "name": "Yuantian Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5aa3795reegxq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07022",
  "fulltext_url": null,
  "id": "5aa3795reegxq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07022",
  "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": "W7220905976",
  "pdf_url": "https://arxiv.org/pdf/2610.07022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5aa3795reegxq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The edge spectral extremal problem for $kK_3$ in nonzero residue classes",
  "title_text": "The edge spectral extremal problem for kK_3 in nonzero residue classes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/5aa3795reegxq",
  "venue": "arXiv",
  "versions": []
}