Isomorphisms of Cyclic Abelian Covers of Symmetric Digraphs, \(III\)

Abstract

Let \(D\) be a connected symmetric digraph, \(\Gamma\) a group of automorphisms of \(D\), and \(A\) a finite abelian group with some specified property. We discuss the number of isomorphism classes of \(g\)-cyclic \(A\)-covers of \(D\) with respect to a group \(\Gamma\) of automorphisms of \(D\). Furthermore, we enumerate the number of \(I\)-isomorphism classes of \(g\)-cyclic \(\mathbb{Z}_{2^m}\)-covers of \(D\) for the cyclic group \(\mathbb{Z}_{2^m}\) of order \(2^m\), where \(I\) is the trivial subgroup of \(Aut(D)\).