Spectral bounds for D(k,q)

Determine a good upper bound on the second-largest eigenvalue λ2(D(k,q)) for k≥7, or determine the spectrum of D(k,q) for k≥5.

Background

The paper states a conjectural universal bound λ2≤2√q for CD(k,q), while complete spectra are known only in limited cases. It asks for stronger eigenvalue bounds or a full spectral determination in the remaining range.

References

Problem 9. Determine a good upper bound on λ2(D(k, q)) for k ≥ 7, or find the spectrum of D(k, q) for k ≥ 5.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Problem 9, Section 6.6