Cubic vertex‐transitive graphs of order 2 pq

A graph is vertex − transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1∉ S and S ={ s −1 | s ∈ S }. The Cayleygraph Cay( G, S ) on G with respect to S is defined as the graph with vertex set G and edge set {{ g, sg } | g ∈ G, s ∈ S }. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4 p or 2 p 2 and in this article we classify all cubic symmetric graphs of order 2 pq , where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2 pq , which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2 pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010