Solution to a conjecture on integral uniform hypercycles

RESUMO

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.