A Classification of Symmetric Graphs of Order 3p

Let Γ be a simple undirected graph and G a subgroup of Aut Γ. Γ is said to be G-symmetric, if G acts transitively on the set of ordered adjacent pairs of vertices of Γ. Γ is said to be symmetric if it is Aut Γ-symmetric. In this paper we give a complete classification for symmetric graphs of order 3p where p is a prime and p > 3. (See the Theorem in the end of Section 4). In the proof of this theorem several consequences of the finite simple group classification, including the classifications of doubly transitive permutation groups and primitive groups of degree mp with p being a prime and m < p, are used.