{
  "abstract": "Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $μ_1\\geq \\cdots \\geq μ_{n-1}\\geq μ_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \\ldots, n$, let $S_k(G)=\\sum_{i=1}^{k}μ_i$. Brouwers conjecture asserts that for any $k\\in\\{1,\\ldots, n\\}$, $S_k(G)\\leq e(G)+\\binom{k+1}{2}$. In [Bounding the sum of the largest Laplacian eigenvalues of graphs, {\\em Discrete Appl. Math.}, 170:95--103, (2014)], Rocha and Trevisan showed that the conjecture holds true for $1 \\leq k \\leq \\lfloor g/5 \\rfloor$, where $g$ denotes the girth of $G$. This bound on $k$ was later improved by Chen in [Improved results on Brouwers conjecture for sum of the Laplacian eigenvalues of a graph, {\\em Linear Algebra Appl.}, 557:327--338, (2018)], who established that the conjecture holds for $1 \\leq k \\leq \\lfloor g/4 \\rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\\leq k\\leq\\left\\lfloor \\frac{m(G)}{2}\\right\\rfloor,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\\geq \\lfloor g/2\\rfloor$, the case constitutes a genuine improvement over the aforementioned results. As an application, we show that if $G$ is a graph of order $n$ and with a perfect matching, then the Brouwers conjecture holds for $1\\leq k\\leq \\left\\lfloor \\frac{n}{4}\\right\\rfloor$. Finally, we provide a new perspective on verifying Brouwers conjecture by proving that $G$ satisfies the Brouwers conjecture if and only if for a fixed positive integer $h$, $\\mathcal{S}_h(\\overline{G})\\leq e(\\overline{G})+\\binom{h+1}{2}$ holds whenever $\\mathcal{S}_h(G)\\leq e(G)+\\binom{h+1}{2}$, where $\\overline{G}$ denotes the complement of $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12673",
  "author_total": 1,
  "authors": [
    {
      "name": "Tahir Shamsher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dw4manhbkm1cf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12673",
  "fulltext_url": null,
  "id": "dw4manhbkm1cf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12673",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7213131162",
  "pdf_url": "https://arxiv.org/pdf/2609.12673",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dw4manhbkm1cf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12673",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12673"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A maximum matching based refinement of Brouwers conjecture",
  "title_text": "A maximum matching based refinement of Brouwers conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/dw4manhbkm1cf",
  "venue": "arXiv",
  "versions": []
}