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<n−1, when n>6. In this article, we confirm this conjecture and prove that for 2≤ k≤ n−1, Cnk is integral if and only if k=n−1 or (n,k)∈{(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.
Related
- Viscosity solution of systems of integral-partial differential equations with interconnected obstacles without Monotonicity Conditions and infinite Lévy measure
- Solution of uniform Turán's Tetrahedron Problem
- Exact Solution of the (2+1)-Dimensional Dunkl-Klein-Gordon Oscillator with a Uniform Magnetic Field in Non-Commutative Phase Space
- A Complete Solution to the Conjecture on the Inverse Collatz–Sinogowitz Irregularity Problem
- AI's solution of Conjecture 9.1
BibTeX
@misc{das2026solution,
title = {{Solution to a conjecture on integral uniform hypercycles}},
author = {Joyentanuj Das and Iswar Mahato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18816},
eprint = {2609.18816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18816},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Solution to a conjecture on integral uniform hypercycles
AU - Joyentanuj Das
AU - Iswar Mahato
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18816
UR - https://arxiv.org/abs/2609.18816
AB - 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≤ k≤ n−1, Cnk is integral if and only if k=n−1 or (n,k)∈{(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.
ER -