Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

TL;DR

This paper classifies and constructs all arc-transitive cyclic and dihedral covers of certain pentavalent symmetric graphs of order twice a prime, providing explicit Cayley graph representations and automorphism groups.

Contribution

It offers a complete classification and explicit construction of such covers, including their automorphism groups, for the specified class of graphs.

Findings

All covers are explicitly constructed as Cayley graphs.

Full automorphism groups of the covers are determined.

Classification covers graphs of order twice a prime.

Abstract

A regular cover of a connected graph is called {\em cyclic} or {\em dihedral} if its transformation group is cyclic or dihedral respectively, and {\em arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.