Comparison of maximal abelian and universal-cover eigenvalues

Determine whether, for the adjacency operator of every finite multigraph, every eigenvalue of the maximal abelian cover is also an eigenvalue of the universal cover, either by proving the inclusion or by constructing a counterexample.

Background

The paper proves that every eigenvalue of the universal cover is also an eigenvalue of the maximal abelian cover. The authors investigate the converse inclusion in the special case of the adjacency operator but leave it unresolved: they neither find a counterexample nor prove the inclusion. Resolving this would clarify the relationship between the two covering constructions' point spectra.

References

As a final remark, despite trying, we could not find a counterexample $G$ to the possibility that when $\mathcal{H}$ is the adjacency operator of $G$, every eigenvalue of $\mathcal{H}{\mathrm{ab}$ is an eigenvalue of $\mathcal{H}{uni}$ --- and neither could we prove it. It would be highly desirable to resolve this matter one way or the other.

— Eigenvalues of Maximal Abelian Covers  (2508.17332 - Li et al., 24 Aug 2025) in Section 1, Introduction