.8 P Order Of 3 Degrees Edge Transitive Graphs

Given a graph X we let V(X), E(X), Arc(X) and A := AutA be the vertex set, the edge set, the arc set and the automorphism group of X respectively. If a subgroup G of AutX" acts transitively on V(X) and E(X), we say that A is G-vertex-transitive and G-edge-transitive respectively. In the special case when G = AutA, we say that A is vertex-transitive and edge-transitive respectively. A regular G-edge- but not G-vertex-transitive graph will be referred to as a G-semisymmetric graph. In particular, if G = AutA the graph is said to be semisymmetric. Moreover, if a graph A has not isolated vertex and AutA acts transitively on Arc(X), we say that A is arc-transitive or symmetric. In this paper, by using the properties of G-semisymmetric cubic graphs and group-theoretic techniques, it is proved that any semisymmetric cubic graph A of order 8p is a regular Zp-cover of Q3, where Zp is normal in AutA, and hence it is vertex-transitive, a contradiction. As a result, we show that any edge-transitive cubic graph of order 8p is symmetric.