The finite vertex-primitive and vertex-biprimitive

A complete classification is given for finite vertex-primitive and vertex-biprimitive -transitive graphs for . The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group , and an infinite family of graphs of valency 5 admitting projective symplectic groups with prime and (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.