Arc-Transitive Pentavalent Graphs of Order Four Times a Prime Power

Abstract

In this paper, we study arc-transitive pentavalent graphs of order \(4p^n\), where \(p\) is a prime and \(n\) is a positive integer. It is proved that no such graph exists for each prime \(p \geq 5\), and all such graphs with \(p = 2\) or \(3\) which are \(G\)-basic (that is, \(G\) has no non-trivial normal subgroup such that the graph is a normal cover of the corresponding normal quotient graph) are determined. Moreover, as an application, arc-transitive pentavalent graphs of order \(4p^2\) and \(4p^3\) are determined.