{
  "abstract": "Graph tensors extend the matrix multiplication tensor by placing a two-party identity tensor on every edge of a graph and keeping one tensor leg at every vertex. We study the asymptotic rank of the tensor associated with the complete graph. For every integer $n\\geq 2$, over the complex numbers, we prove \\[ \\left\\lfloor\\frac{n^2}{4}\\right\\rfloor \\leq \\log_2\\widetilde R\\bigl(\\mathsf{GT}_2(K_n)\\bigr) \\leq \\frac{n^2}{4}. \\] The upper bound matches the maximum-cut lower bound for every even $n$ and differs from it by only $1/4$ for every odd $n$. In particular, the exponent per edge is below $2/3$ for every $n\\geq5$, answering a question of Christandl, Vrana, and Zuiddam. Our proof extends the polynomial multiplication approach underlying OpenAI's bound $ω\\leq9/4$. The main step separates any fixed number of shared tensor legs simultaneously, using only linearly many direct-sum copies in the number of blocks. This yields an entropy inequality for higher-order tensors. We apply it to the tensor for multiplying $n-1$ polynomials, whose rank is linear in the input length. Discrete concavity and a shifted tripling inequality then force a lower bound on a symmetrized spectral profile. Comparing this growth with polynomial interpolation gives the stated upper bound while preserving all $n$ tensor legs. We also prove an operational barrier for these higher-order polynomial constructions. For $n\\geq3$, the specified formal reduction rules cannot certify an exponent below $n^2/4$ for $\\mathsf{GT}_2(K_n)$, even after taking tensor powers and allowing subexponential overhead and common auxiliary factors. This extends the symmetric three-leg barrier of Amanov, van den Berg, and Zuiddam and explains the remaining gap for odd complete graphs within this method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34273",
  "author_total": 1,
  "authors": [
    {
      "name": "Huanyi Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fmsqcp4v2mzdv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34273",
  "fulltext_url": null,
  "id": "fmsqcp4v2mzdv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34273",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223935065",
  "pdf_url": "https://arxiv.org/pdf/2610.34273",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fmsqcp4v2mzdv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34273",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34273"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotic Rank of Complete Graph Tensors",
  "title_text": "Asymptotic Rank of Complete Graph Tensors",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/fmsqcp4v2mzdv",
  "venue": "arXiv",
  "versions": []
}