Symmetric Graph On Valency 4 And Semisymmetric Graph

This paper studies the classification of the Cayley graph ? which is teravalent one-transitive but not one-regular and the construction of semisymmetric graph.Tutte proved a cubic graph must be at most five-arc-transitive in 1947.Then scholars started put their attention on the classification and construction of smal-I valency s-arc-transitive graph,and the investigation gradually became a top-ic question.In chapter three,we study the classification of the Cayley graph? which is teravalent one-transitive but not one-regular and give it an exactly characterization.Let ? be a teravalent X-arc-transitive Cayley graph of dihedral group.let X ? Aut(?)and Xv be the stabilizer of X on v?V(?)We give r an exactly characterization.When |Xv|?24,we give a compete classification of Xv:D8,SmallGroup(16,3),D16,SD16,D8×Z2.And ? is isomorphic to the Oc-tahedron graph,complete bipartite graph K 4 4,W(5,2)or W(6,2)when the order of Xv is no more than 24 and G is core-free in X.In particular,if X = Aut(?),then? is only isomorphic to the Octahedron graph.A simple undirected regular graph ? is said to be semisymmetric,if the action of Aut(?)on E(?)is transitive and the action of Aut(?)on V(?)is not transitive.It is easy to know taht a semisymmetric graph must be semitransitive and bipartite with two parts of equal size.While we construct semisymmetric graph,we face a big problem:a regular edge-transitive graph also is usually a vertex-transitive graph.In 1976,Folkman first systematically studied semisymmetric graphs,he constructed some examples include the smallest semisymmetric graph.In chapter four,we focus on the construction of semisymmetric graph.We use finite groups and bi-coset graph to construct semisymmetric graph.With the new method,we also give an infinite family of such graph:?'=?(q,p),? is a bi-coset graph andV(?')= {(u,i)| u?U,i?Z_q}?{(w,i)| w? W,i ? Z_p?,E(?')= {{(u,i),(w,j)} | ?u,w} ? E(?),i ? Z_q,j ? Z_p?.