On Vertex-Uniprimitive Non-Cayley Graphs of Order pq

Let p and q be distinct odd primes. Let Γ = (V (Γ),E(Γ)) be a non-Cayley vertex-transitive graph of order pq. Let G ≤ Aut(Γ) acts primitively on the vertex set V (Γ). In this paper, we show that G is uniprimitive which is primitive but not 2-transitive and we obtain some information about p,q and the minimality of the Socle T = Soc(G)