Locally Primitive Graphs On Some Groups

Let Γ be a vertex transitive graph and Aut(F) the full automorphism group and X≤Aut(Γ), for a vertex v, a graph F is called X-locally primitive if the stabilizer Xv acts primitively on Γ(v). A graph F is called (X, s)-arc transitive if X acts transitively on the set of all s-arcs of Γ. Interest in such graphs goes back to Tutte’s seminal work showing that there exist no (X, s)-arc transitive graphs of valency three satisfying s≥6.A (X,2)-arc transitive graph is X-vertex transitive and X-locally primitive. There are some typical examples of locally primitive graphs: n cycle Cn, the complete graphs Kn, the complete bipartite graphs Kn,n and Kn,n-nK2, the graph obtained by deleting a l-actor from Kn,n. In fact, these graphs are2-arc transitive.The classes of2-arc transitive graphs and locally primitive graphs have been extensively studied. The main result of the thesis is classifying the undirected vertex transitive locally primitive graphs:Firstly, the thesis gives a complete classification of X-locally prim- itive graphs with order p2qr of group X.Secondly, the thesis completely classifies J1-locally primitive graphs.Thirdly, the thesis gives a classification of J2-locally primitive graphs.