Spectral-gap conjecture for maximal abelian covers of regular graphs

Establish the conjecture that maximal abelian covers of regular graphs have no spectral gaps.

Background

The paper confirms one of the principal conjectures of Higuchi and Nomura by proving that maximal abelian covers of regular multigraphs have no eigenvalues. It identifies the remaining conjecture as the absence of spectral gaps, a question concerning the spectral structure of maximal abelian covers that had previously been established only for certain classes of regular graphs.

References

Theorem~\ref{thm:regular} is the second result of this paper, and is our main contribution. It confirms one of the main conjectures about the spectral structure of maximal abelian covers of regular graphs Conj. 6.12, the remaining one being the lack of spectral gaps .

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