Vertex-Transitive Graphs Of Some Groups With Even Order

We are mainly interested in the symmetry of graphs in studying groups and graphs. The symmetry of graphs is usually described by certain transitivity of automorphism groups of the graphs. The Cayley graph and Sabidussi coset graph are the classical representativities of these graphs. The main purpose of this thesis is studying the CI property and normality of Cayley graphs and coset graphs of groups with order 4p and 6p.This thesis first study the CI property of connected Cayley graphs of valency 3 of dihedral groups with order 4p, and give their complete classification. At the same time, We prove that the dihedral groups with order 4p is not weak 3-CI groups, so negative the guess that all finite groups are weak 3-CI groups, which is given by C. H. Li in 1998(refer to [1]).Then this thesis resolve the normality of the directed Cayley graphs of valency 2 of groups with order 6p, and prove a sufficient and necessary condition that the directed Cayley graphs of valency 2 of groups with order 6p is not normal.Finally, since the reseaching of the symmetry of Sabidussi coset graphs has more general significance, because very vertex-transitive graph is always some coset graph of its full automorphism group,So, we also define the normality of coset graphs like one of Cayley graphs, and prove that the connected coset graphs of valency 3 of dihedral groups with order 6p is normal in most instances. The method used in this thesis is mainly group-theoretic. For concepts of group theory and algebraic graph theory we refer the reader to [2, 3, 4].