Enumerating regular graph coverings whose covering transformation groups are \(\mathbb {Z}_p\)-extensions of a cyclic group

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by Kwak and Lee. For regular graph coverings, Kwak, Chun and Lee enumerated the isomorphism classes of graph coverings when the covering transformation group is a finite abelian or a dihedral group in Kwak et al. (SIAM J Discrete Math 11:273–285, 1998). In 2018, the isomorphism classes of graph coverings are enumerated when the covering transformation groups are \(\mathbb {Z}_2\) ${\mathbb{Z}}_2$ -extensions of a cyclic group. As a continuation of this work, we enumerate the isomorphism classes of coverings of a graph when the covering transformation groups are \(\mathbb {Z}_p\) ${\mathbb{Z}}_p$ -extensions of a cyclic group for an odd prime integer p.