{
  "abstract": "A hypergraph is said to be integral if all of its adjacency eigenvalues are integers. Recently, Portugal and Del-Vecchio in [\\emph{Appl. Math. Comput.} 504: 129507 (2025)] studied the integral hypergraphs and gave a characterization of integral hypercycles in three particular cases: $3$-uniform, $4$-uniform and $5$-uniform hypercycles. In the same article, they conjectured that the $k$-uniform hypercycle on $n$ vertices $\\Cnk$ is never integral for $k<n-1$, when $n>6$. In this article, we confirm this conjecture and prove that for $2\\le k\\le n-1$, $\\Cnk$ is integral if and only if $k=n-1$ or $(n,k)\\in\\{(4,2),(6,2),(6,3),(6,4)\\}$. The proof begins with computing the complete adjacency spectrum of $\\Cnk$ and then uses Niven's theorem, a cyclotomic-unit lemma, an elementary property of Euler's totient function, and the Galois symmetry of cyclotomic fields to complete it. Our result gives a complete characterization of $k$-uniform integral hypercycles on $n$ vertices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18816",
  "author_total": 2,
  "authors": [
    {
      "name": "Joyentanuj Das"
    },
    {
      "name": "Iswar Mahato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eycp699cafat8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18816",
  "fulltext_url": null,
  "id": "eycp699cafat8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18816",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7213530728",
  "pdf_url": "https://arxiv.org/pdf/2609.18816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eycp699cafat8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Solution to a conjecture on integral uniform hypercycles",
  "title_text": "Solution to a conjecture on integral uniform hypercycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/eycp699cafat8",
  "venue": "arXiv",
  "versions": []
}