Kida's formula for graphs with ramifications

Recently Iwasawa theory for graphs is developing. A significant achievement includes an analogue of Iwasawa class number formula, which describes the asymptotic growth of the numbers of spanning trees for ${\mathbb{Z}}_p$-coverings of graphs. Moreover, an analogue of Kida’s formula concerning the behavior of the $\lambda$- and $\mu$-invariants is obtained for unramified coverings. In this paper, we establish Kida’s formula for possibly ramified coverings.