Symmetric Graphs of Order a Product of Two Distinct Primes

A simple undirected graph Γ is said to be symmetric if its automorphism group Aut Γ is transitive on the set of ordered pairs of adjacent vertices of Γ, and Γ is said to be imprimitive if Aut Γ acts imprimitively on the vertices of Γ. Let k and p be distinct primes with k < p. This paper gives a classification of all imprimitive symmetric graphs on kp vertices for k ≥ 5. The cases k < 5 have been treated previously by Cheng and Oxley (k = 2) and the second and third authors (k = 3), and the classification of primitive symmetric graphs on kp vertices with k ≥ 5 was done by the first and third authors.