Finite Edge-Transitive Graphs

This present thesis is a contribution to the application of group theory to graph theory.For a graphΓ=(V,E), the number of vertices is called the order ofΓ. For a subgroup G of AutΓ, the graph F is called G-vertex-transitive or G-edge-transitive if G acts transitively on the vertex set V or the edge set E, respectively. An arc of F is an ordered adjacent pair of vertices and F is called G-arc-transitive if G acts transitively on the set of arcs ofΓ. A regular graph F that is G-edge-transitive but not G-vertex-transitive is called a G-semisymmetric graph. AndΓis called G-half-transitive if it is G-vertex-transitive and G-edge-transitive but not G-arc-transitive.It is well-known that ifΓis a regular G-edge-transitive graph, then exactly one of the following occurs:(1) F is arc-transitive; (2)Γis G-half-transitive; or (3)Γis G-semisymmetric. Each of the above three families of edge-transitive graphs has been extensively studied in the last few decades. Thus it is meaningful to characterize and classify edge-transitive graphs.Our main work in this thesis is to investigate edge-transitive graphs of square-free order. In the literature, the problem of characterizing and classifying edge-transitive graphs of square-free order has received considerable attentions, and so does the vertex-transitive case. This problem for lots of special cases has been solved, many of which depended on the Liebeck-Saxl's classification. The Li-Seress's classification of primitive permutation groups with square-free degree given by Li and A. Seress in 2004 provided a more important tool for our research.At first, we investigated the edge-transitive basic graphs of square-free order. A memberΓis called basic if each of its nontrivial normal quotients has at most two vertices. It is shown that each edge-transitive graph of square-free order is a normal cover of a basic member or its quotient is a star. Thus, basic members form the core of edge-transitive graphs of square-free order. For graphs with certain valency, we prove that except for a few special families of graphs, only finitely many edge-transitive members are basic. A complete classification of edge-transitive graphs which also vertex-transitive with valency 4 is given in this thesis. It is isomorphic to one of the following:Cn[(?)2], arc-regular metacirculant, edge-regular metacirculant or cyclic cover of finitely members.Based on the characterization of edge-transitive basic graphs, we further studied the locally primitive graphs of square-free order. A graphΓis called G-locally primitive if Gαacts primitively onΓ(a) for every vertex a. It is shown that the arc-transitive members with certain valency consist of a few well char-acterized families of graphs (ie, normal dicirculants, PSL(2,p)-locally primitive graphs and PSL(2, p)-edge-transitive graphs of valency 4), and normal covers of finitely many graphs admitting an alternating group, a sporadic group or a group of Lie type. Further, the members of locally primitive arc-transitive graphs with valency less than 8 are classified. They contain normal dicirculants of prime va-lency, PSL(2,p)-locally primitive graphs and finitely 2-arc-transitive graphs given in this thesis.The above classifications allow us naturally to study 2-arc-transitive graphs of square-free order. AΓis called a (G,2)-arc-transitive graph if G is transitive on VΓand on the set of 2-arcs ofΓ. We classified 2-arc-transitive graphs of square-free order admitting alternating or symmetric groups in this thesis. We shall study general 2-arc-transitive graphs in future.In the process of our study, there exists a class of G-edge-transitive graphs of square-free with valency 4 such that G has a normal subgroup M which is semiregular and has exactly two orbits on V. We extended this class of graphs to the general case and proved that every bi-normal Cayley graph is not 3-arc-transitive, it gives an negative answer to the question posed by Li in [62].