Vertex-Primitive Graphs of Order a Product of Two Distinct Primes

Let k and p be odd primes with k < p. All vertex-primitive graphs of order kp are classified, and those which are symmetric, or are edge-transitive but not symmetric. or are not Cayley graphs are identified. In addition, a classification is given of all vertex-primitive, edge-transitive lantisymmetric) directed graphs or order kp. The classification of the vertex-primitive symmetric graphs of order kp is used by the authors and others to complete the classification of symmetric graphs or order kp. The classification of the vertex-primitive non-Cayley graphs of order kp is used by B.D. McKay and the first author in an investigation of vertex-transitive non-Cayley graphs.