This paper studies several problems in voltage graph theory (useful in constructing graph imbeddings), such as equivalence and regularity of coverings generated by (permutation) voltage graphs, and automorphism groups. To consider a sample result, let \(\chi\) be either an ordinary voltage assignment (OVA) or a permutation voltage assignment (PVA), from a group G (G is a symmetric group in the permutation case) to the arcs of a connected pseudograph K having spanning tree T and designated vertex v. For any \(e\not\in T\), there is an oriented closed walk \(\alpha_ e\) in K, based at v, such that: (i) \(\alpha_ e-e\subseteq T\); (ii) the orientation of \(\alpha_ e\) and e agree; (iii) \(\alpha_ e\) has minimum length subject to (i) and (ii). Now let \(\chi '(e)=\chi *(\alpha_ e)\)- the voltage product for \(\alpha_ e\)- if \(e\not\in T\), and \(\chi '(e)=1\) otherwise. The new voltage assignment \(\chi\) ' for K is called the (T,v)- reduction of \(\chi\). Theorem. Let \(\chi\) and \(\lambda\) be two OVA's (PVA's) on K, both in the same group \(G(S_ n)\). Let \(\chi\) ' and \(\lambda\) ' be the corresponding (T,v)-reductions. Then the coverings \(P: K^{\chi}\to K\) and \(q: K^{\lambda}\to K\) are equivalent if and only if there is an automorphism (inner automorphism) of \(G(S_ n)\) such that \(A\circ \chi '=\lambda '\).