.4 Pq And Order 3 Degrees Connected Semi-symmetrical Map,

Given a graph X we let V(X),E(X),Arc(X) and A = Aut(X) be the vertex set,the edge set,the arc set and the automorphism group of X respectively. If a subgroup G- of Aut(X) acts transitively on V(X) and E(X) ,we say that X is G-vertex-transitive and G-edge-transitive respectively. In the special case when G = Aut(X\ we say that X is vertex-transitive and edge-transitive respectively. A regular G-vertex but not G-vertex-tranxitive graph will be refferred to as a G-semisymmetric graph. In particular, if G = Aut(X) the grahp is said to be semisymmetric. Moreover if a graph X is not isolate vertex and G = Aut(X) acts transitively on Arc(X], we say that X is arc-transtive or symmetric. In this paper, by using the properties of G-semisymmetric cubic graphs and group-theoretic techniques, it is proved that all connected cubic edge-transitive graph of order 4pq are symmetric (p and q are different odd primes and p, q > 3), when their full automorphism contains no unsolved minimum normal subgroup, furthermrore writer constructs connected cubic semisymmetric graph of order 4pq, when their full automorphism contains unsolved minimum normal subgroup.