On minimal symmetric automorphism groups of finite symmetric graphs

Let r be finite connected and G a group of automorphisms of r which is transitive on vertices. Suppose that, for a vertex 0 of r, S ~ G~(O') ::; Aut S for some simple group S with S acting primitively on the set r( a) of neighbours of 0, and suppose that G is minimal with these properties. Then one of: (i) G is a nonabelian simple group, (ii) r is a Cayley graph for a normal subgroup N of G and G = N.S, (iii) r is bipartite, (iv) r is a proper cover of a graph of the same valency and with the same properties. In the special case where r has prime valency this is result of Lorimer. More details of the structure of rand G are obtained for graphs which satisfy (ii) or (iii) but are not proper covers as in (1V). Constructions are given for several families of examples.