Isomorphisms of Connected Cyclic Abelian Covers of Symmetric Digraphs

Abstract

Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group, \(g \in A\) and \(\Gamma\) a group of automorphisms of \(D\). We consider the number of \(T\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. Specifically, we enumerate the number of \(I\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order and the trivial automorphism group \(\Gamma\) of \(D\), when \(A\) is the cyclic group \({Z}_{p^n}\) and the direct sum of \(m\) copies of \({Z}_p\) for any prime number \(p (> 2)\).