Local 2 - Arc Pass Complete Bipartite Graph

Letгbe a graph, and let Autгdenote its full automorphism group. A graphгis called (G,2)-arc transitive if G is transitive on the set of 2-arcs ofг, and a graphгis called locally (G,2)—arc transitive if for each vertex v, Gv acts transitively on 2—arcs starting at v.Characterizing and constructing locally 2—arc transitive graphs have received much attention on the literature, see, for example, finite locally s—arc transitive graphs and locally s—arc transitive graphs with two different quasiprimitive actions have been characterized by Li and Prager. For more results, see the references [5,6,7].Let F be a graph, it is called a complete bipartite graph if it with biparts U and W such that |U|=m,|W|= n and each vertex in U is adjacent to all vertices in W, but each vertex in U is not adjacent to any other vertex inside of U, which is denoted by Km,n. The automorphism group ofгis Aut(Km,n)= Sm x Sn. In particular, if m= n, then Aut{Kn.n)= Sn IS2. In general, we consider the biparts as two orbits of groups acting.The study about edge-transitive action on the complete bipartite graphs is a important problem, which involves graph theory, geometry and group theory. The main purpose of this paper is to character a specular case, which is proposed by the mathematician Leemans in Belgium, that is, to character the finite groups that act on a complete bipartite edge-transitive graph and to obtain a classification of the automorphism groups for the locally 2—arc transitive case.