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.