{"ID":22964820,"CreatedAt":"2026-09-17T02:29:35.102872001Z","UpdatedAt":"2026-09-17T02:29:35.102872001Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18816","arxiv_id":"2609.18816","title":"Solution to a conjecture on integral uniform hypercycles","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 $k\u003cn-1$, when $n\u003e6$. 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.","short_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 $5$-uni...","url_abs":"https://arxiv.org/abs/2609.18816","url_pdf":"https://arxiv.org/pdf/2609.18816v1","authors":"[\"Joyentanuj Das\",\"Iswar Mahato\"]","published":"2026-09-16T15:23:55Z","proceeding":"math.CO","tasks":"[\"math.CO\"]","methods":"[]","has_code":false}