{
  "abstract": "We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of $5/8$ for Quantum Max Cut and $5/7$ for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of $0.614$ for Quantum Max Cut and $0.674$ for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-$k$ eigenvalues of a Graph Laplacian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.14994",
  "author_total": 4,
  "authors": [
    {
      "name": "Ainesh Bakshi"
    },
    {
      "name": "Arpon Basu"
    },
    {
      "name": "Pravesh Kothari"
    },
    {
      "name": "Anqi Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4jhwch0e58fpb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.14994",
  "fulltext_url": null,
  "id": "4jhwch0e58fpb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.14994",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7161249103",
  "pdf_url": "https://arxiv.org/pdf/2605.14994",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4jhwch0e58fpb.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:20:55Z",
      "native_id": "2605.14994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.14994"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Sharp Bounds on the Eigenvalues of Kikuchi Graphs and Applications to Quantum Max Cut",
  "title_text": "Sharp Bounds on the Eigenvalues of Kikuchi Graphs and Applications to Quantum Max Cut",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/4jhwch0e58fpb",
  "venue": "arXiv",
  "versions": []
}