{
  "abstract": "Let $S_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of a connected graph $G$ of order $n$ and size $m$. Write $\\mathrm{PA}_{n,ω}$ for the graph obtained from an $ω$-vertex clique by attaching $n-ω$ pendant vertices to one of its vertices, and set \\[ M_{n,k}:=\\binom{k+1}{2}+n-k-1, \\] the number of edges of $\\mathrm{PA}_{n,k+1}$. For $n/2<k\\le n-2$, we prove the sharp bound \\[ S_k(G)\\le \\frac{2k}{n}m+ \\frac{2(n-k)}{n}M_{n,k}-(n-k-1), \\] with equality attained by $\\mathrm{PA}_{n,k+1}$. This bound is complementary to Brouwer's inequality and is strictly stronger when $m<M_{n,k}$. Combining our bound with Brouwer's inequality, we resolve and strengthen a conjecture of Vinagre, Del-Vecchio, Justo, and Trevisan: for every $n$, the pineapple $\\mathrm{PA}_{n,1+\\lfloor2n/3\\rfloor}$ maximizes the Laplacian energy among all connected graphs of order $n$; moreover, for $n>4$, it is the unique maximizer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06442",
  "author_total": 1,
  "authors": [
    {
      "name": "Seyed Ahmad Mojallal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4afn7503904q8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06442",
  "fulltext_url": null,
  "id": "4afn7503904q8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06442",
  "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": "W7212038607",
  "pdf_url": "https://arxiv.org/pdf/2609.06442",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4afn7503904q8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06442",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06442"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound",
  "title_text": "A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4afn7503904q8",
  "venue": "arXiv",
  "versions": []
}