An Iwasawa-type asymptotic formula for multiple ${\mathbb{Z}}_p$-coverings of graphs

For a possibly ramified ${\mathbb{Z}}_p^d$-covering of connected graphs, we establish an Iwasawa-type asymptotic formula for the growth of the $p$-adic valuations of the complexities. The formula is expressed as a polynomial in $n$ and $p^n$ with explicit leading coefficients $\lambda$ and $\mu$; in particular, we eliminate the error term of the form $O(p^{(d-1)n})$ appearing in earlier work. We then establish a Kida-type formula describing the behavior of $\lambda$ under a $p$-covering between ${\mathbb{Z}}_p^d$-coverings, assuming $\mu =0$. Finally, for any fixed $p$ and integer $d\ge 2$, we construct an unramified ${\mathbb{Z}}_p^d$-covering of a bouquet with prescribed $\lambda$- and $\mu$-invariants.