Portugal–Del-Vecchio integral-hypercycle conjecture

Let $\mathcal{C}_n^k$ denote the $k$ -uniform hypercycle on $n$ vertices. A hypergraph is called integral if all of its adjacency eigenvalues are integers. The conjecture states that, for $n>6$ and $2\le k<n-1$ , the hypercycle $\mathcal{C}_n^k$ is not integral.

References

Primary source

arXiv

Additional references

Solution to a conjecture on integral uniform hypercycles — arXiv — Joyentanuj Das, Iswar Mahato

algebraic-combinatorics spectral-graph-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new preprint claims to settle the conjecture completely, but no independent check is reported.

The conjecture seeks an all-parameter characterization of when the relevant hypercycle is integral, extending classifications for small uniformities.

Recent preprint

Joyentanuj Das and Iswar Mahato claim that integrality occurs exactly when $k=n-1$ or for four exceptional parameter pairs. This would resolve the conjecture completely, but the supplied evidence contains no independent verification.

Current status (as of September 2026): A complete characterization is claimed, with integrality exactly for $k=n-1$ or four exceptional parameter pairs, but the claim remains unverified.

Sources

No solutions have been posted yet.