Defect constancy in constant $Z_p$-towers of strongly connected digraphs

Establish that the defect is constant throughout every constant $Z_p$-tower of strongly connected digraphs: for a constant $Z_p$-tower $(X_m)_m$ of strongly connected digraphs, prove that $delta(X_m)$ is independent of $m$.

Background

The paper studies the defect delta(X)delta(X) of a finite digraph, defined as the difference between the algebraic and geometric multiplicities of $1$ as an eigenvalue of its adjacency matrix. The authors investigate this invariant along constant ZpZ_p-towers, obtained from constant voltage assignments and successive cyclic covers of a base digraph.

The unresolved conjecture concerns arbitrary constant ZpZ_p-towers of strongly connected digraphs. The paper proves the asserted constancy for several families of Cayley graphs, including Cayley graphs of finite abelian groups, dihedral groups, and groups isomorphic to Z/pZtimesZ/(p−1)ZZ/pZ times Z/(p-1)Z, but does not establish it for all strongly connected digraphs.

References

In particular, a main focus of this article is to provide evidence towards the following conjecture of Lei--Müller Conjecture~7.11. For precise definitions of terms used in the following conjecture, we refer the reader to Sections~\ref{Iwasawa theory} and \ref{Bowen Franks}. Let $(X_m)_m$ be a constant $Z_p$-tower of strongly connected digraphs. The defect $\delta$ is constant throughout the $Z_p$-tower.

— Iwasawa theory of (directe) Cayley graphs  (2609.16702 - Kundu et al., 15 Sep 2026) in Section 1, Introduction, immediately before Section 1.1 (Main Results)