Flat-band absence for maximal abelian covers of biregular graphs

Establish whether the maximal abelian cover of every finite biregular multigraph has no eigenvalues other than zero when the vertex potentials vanish.

Background

The paper proves that the maximal abelian cover of every finite regular multigraph has no eigenvalues, and extends this conclusion to many biregular graphs under the assumption that all vertex potentials are zero. The authors explicitly state that the general biregular case remains unresolved, making it a natural continuation of their methods.

References

Furthermore, by using the ideas of the proof of Theorem \ref{thm:regular} we were able to prove $\mathcal{H}{ab}$ has no eigenvalues other than zero for many bi-regular graphs (for vertex potentials equal to zero); the bi-regular case in general remains open and is a natural follow on question from our paper.

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