On the Iwasawa theory of Cayley graphs

This paper explores Iwasawa theory from a graph-theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within \(\mathbb {Z}_\ell \) ${\mathbb{Z}}_{\ell }$ -towers of Cayley graphs, revealing connections between graph theory, number theory, and group theory. Key results include the factorization of associated Iwasawa polynomials and the decomposition of \(\mu \) $\mu$ - and \(\lambda \) $\lambda$ -invariants. Additionally, we apply these insights to complete graphs, establishing conditions under which these invariants vanish.