Isomorphisms of some graph coverings (Q1322193)

Let \(G\) be a simple graph, \(\Gamma\) a group of automorphisms of \(G\), and \(F= \text{GF}(p)\). Let \(A(G)\) be the arc set of the corresponding symmetric digraph of \(G\). Voltage assignment \(\alpha\) on \(G\) is a function \(\alpha: A(G)\to F\) such that \(\alpha((x,y))= - \alpha((y,x))\). Let \(\alpha\) be the voltage assignment on \(G\) over \(F\) and \(G^ \alpha\) the derived graph with vertices \(V(G^ \alpha)= V(G)\times F\) and \(((x,i),(y,j))\in A(G^ \alpha)\) if and only if \((y,x)\in A(G)\) and \(j= \alpha(x,y)+ i\). Natural projections \(p_ \alpha\), \(p_ \beta\) are called \(\Gamma\)- isomorphic if there exists an isomorphism \(\psi: G^ \alpha\to G^ \beta\) and \(g\in \Gamma\) such that \(p_ \beta\psi= gp_ \alpha\). The author enumerates the number of \(\Gamma\)-isomorphism classes of the derived graph coverings of \(G\) with voltages in a finite field of prime order \(p\) for \(p>2\).