{
  "abstract": "For every $n$ and $r$ with $1\\le r$ and $\\binom{r+1}{2}\\le n$, we construct a connected unweighted graph on $n$ vertices whose Max-Cut semidefinite relaxation has a unique optimal solution of rank $r$. The main examples are the Kneser graphs $K(m,2)$, whose unique optima have rank $m-1$. A six-vertex wheel supplies rank three, and attaching pendant vertices preserves both rank and uniqueness. For dense random graphs, we also prove a polynomial lower bound on the optimal rank.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34324",
  "author_total": 2,
  "authors": [
    {
      "name": "Avinash Bhardwaj"
    },
    {
      "name": "Vishnu Narayanan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdzda9em58pmc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34324",
  "fulltext_url": null,
  "id": "fdzda9em58pmc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34324",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7223996004",
  "pdf_url": "https://arxiv.org/pdf/2610.34324",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdzda9em58pmc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34324",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34324"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal solution ranks in the Max-Cut semidefinite relaxation",
  "title_text": "Optimal solution ranks in the Max-Cut semidefinite relaxation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fdzda9em58pmc",
  "venue": "arXiv",
  "versions": []
}