Isomorphisms of Cyclic Abelian Covers of Symmetric Digraphs

Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group with some specified property and \(g \in A\). We present a characterization for two \(g\)-cyclic \(A\)-covers of \(D\) to be isomorphic with respect to a group \(\Gamma\) of automorphisms of \(D\), for any \(g\) of odd order. Furthermore, we consider the number of \(\Gamma\)-isomorphism classes of \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. We enumerate the number of isomorphism classes of \(g\)-cyclic \({Z}_{p^n}\)-covers of \(D\) with respect to the trivial group of automorphisms of \(D\), for any prime \(p (> 2)\), where \(\mathbb{Z}_{p^n}\) is the cyclic group of order \(p^n\). Finally, we count \(\Gamma\)-isomorphism classes of cyclic \({F}_p\)-covers of \(D\).