Let G be a simple undirected graph and G a subgroup of the automorphism group AutG. Then G is said to be G-symmetric, and G is said to be symmetric on G, if G is transitive on the set of ordered pairs of adjacent vertices of G; G is said to be symmetric if AutG is symmetric. In this paper we give a complete classification for vertex-primitive symmetric graphs of order 6p where p is a prime. The main theorem of this paper is the following:
Theorem 1 If G is a G-symmetric graph of order 6p where p is a prime,and if Gacts on V(G) primitively, then G is one of the following: 6pK1, K6p, T12, T12c, T13, T13c, (M11)6615, (M11)6620, (M11)6630, L2(17)1023, L2(17)1026, L2(17)1028, L2(17)10224, L2(17)10224¢, L2(17)10224¢¢, L2(13)787, L2(13)787(1), L2(13)7814, L2(13)7814(1), [`L]2(13)787(i), 0 £ i £ 2, [`L]2(13)7814, [` L]¢2(13)7814, [`L]2(13)7828
In the proof of this theorem several consequences of the classifications of vertex-primitive graphs of degree mp with p in , are used. The group- graph-theoretic notation and terminology used in this paper are standard in general; the reader can refer to  when necessary.
1 This work was supported by the National Natural Science Foundation of China (19901012).