  Let $p$ be a rational prime, and let $X$ be a connected finite graph. In this
article we study voltage covers $X_\infty$ of $X$ attached to a voltage
assignment ${\alpha}$ which takes values in some uniform $p$-adic Lie group
$G$. We formulate and prove an Iwasawa main conjecture for the projective limit
of the Picard groups $\text{Pic}(X_n)$ of the intermediate voltage covers
$X_n$, ${n \in \mathbb{N}}$, and we prove one inclusion of a main conjecture
for the projective limit of the Jacobians $J(X_n)$.
  Moreover, we study the $\mathfrak{M}_H(G)$-property of
$\mathbb{Z}_p[[G]]$-modules and prove a necessary condition for this property
which involves the $\mu$-invariants of $\mathbb{Z}_p$-subcovers ${Y \subseteq
X_\infty}$ of $X$. If the dimension of $G$ is equal to 2, then this condition
is also sufficient.
