Classification of vertex-transitive graphs of order a prime cubed — I Original Research Article

Marus̆ic̆ has shown that every vertex-transitive graph of order p3 is isomorphic to a Cayley graph. Hence to classify vertex-transitive graphs of order p3, it suffices to determine necessary and sufficient conditions for two Cayley graphs of a group G of order p3 to be isomorphic (the isomorphism problem). In this paper, we determine necessary and sufficient conditions for two Cayley graphs of Zp×Zp2 to be isomorphic. In subsequent papers, we will consider the remaining groups of order p3 for which the isomorphism problem has not been solved.