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Let $p$ be a rational prime, and let $X$ be a connected finite graph. In thisarticle we study voltage covers $X_\infty$ of $X$ attached to a voltageassignment ${\alpha}$ which takes values in some uniform $p$-adic Lie group$G$. We formulate and prove an Iwasawa main conjecture for the projective limitof 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 conjecturefor 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 propertywhich involves the $\mu$-invariants of $\mathbb{Z}_p$-subcovers ${Y \subseteqX_\infty}$ of $X$. If the dimension of $G$ is equal to 2, then this conditionis also sufficient.

On the non-commutative Iwasawa main conjecture for voltage covers of graphs