Some Notes On Symmetric Graphs

For a given finite connected graphΓ, let V(Γ),E(Γ),A(Γ) and Aut(Γ) denote the vertex set, the edge set, the arc set and the automorphism group ofΓ,respectively. For anyν∈V(Γ), we use N(ν) to denote the neighborhood ofνinΓ. This paper discusses the lifting problem of the graph K₆ with the automorphism group S₆, and we find that there are no connected regular covering graphs (?) of K₆ satisfying the following two properties:(1)the covering transformation group is isomorphic to the elementary abelian group Z₂⁴, and (2) the group of fibre-preserving automorphisms of a covering graph acts 2-arc-transitively.LetΓbe a simple directed (or undirected) graph and G be a subgroup of automorphism group Aut(F) ofΓ, thenΓis said to be G-vertex transitive, Gvertex primitive, G-edge transitive, or G-symmetric if G is transitive on the vertex set ofΓ, primitive on vertex set ofΓ, transitive on the edge set, or transitive on the ordered pairs of adjacent vertices ofΓ, respectively. IfΓhas one of these properties for G=Aut(Γ), the prefix Aut(Γ) is often omitted andΓis said to be vertex transitive, vertex primitive, edge transirive, or symmetric, resperctively. The number of the vertices ofΓis called the order ofΓ. Praeger and Xu determined all vertex primitive graphs and digraphs of order a product of two distinct primes. For the case where k is not a prime, even for k=4 or 6, the problem of classifying imprimitive symmetric graphs of order kp is very difficult; a group of mathematicians are working on it.Some works about the classification of symmetric graphs of order 6p were done, Wang Ruji gives a classification for solvable symmetric graphs of 6p, and Guo Dachang give a classification for symmetric graphs of order 30 which is unique case of p＜6. The aim of this paper is to classify the vertex-primitive symmetric graphs of order 6p. These work were essentially done in [11], however in [11] there is no such situation: G=PSL(2,13) acting on the setΩof cosets of subgroup H≌D₁₄. Then m=|Ω|=78=6p, G has rank 9, and the suborbits of G have one of length 1, five of length 7, three of length 14. Thus, this paper gives a completed list of symmetric graphs of order 6p.