Classifying Pentavalent Symmetric Graphs of Order \(40p\)

Abstract

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order \(40p\) for each prime \(p\). It is shown that a connected pentavalent symmetric graph of order \(40p\) exists if and only if \(p = 3\), and up to isomorphism, there are only two such graphs.