The Normality Of Cayley Graphs And 1-Regular Cayley Graphs

The symmetry of graphs has been a very hot issue problem in the study of groups and graphs, and it mainly depends on some transitive properties of act-ing by the automorphism groups of the graphs to describe. The Cayley graph is a classical representatives for the symmetry graphs because of the simple construct-ing, high degree of symmetry and diversity of species. The symmetry of Cayley graphs depends on the information how deeply we know from full automorphism groups of them and their normalities are a fundamental problem for that.So we study the normalities of the Cayley graphs in the chapter 3 and we obtain the follow results:we get three sufficient conditions for non-normal Cayley graphs and by using the conditions, four infinite families of connected non-normal Cayley graphs of valency 6 on A5 are determined, which generalizes a result about the normalities of Cayley graphs of valency 3 or 4 on A5.A graph F is called 1-regular if its full automorphism group Aut(F) acts regularly on its arcs. The 1-regular graph is the main object in the study of groups and graphs also. But Recently people are mainly discussing how to construct the 1-regular graph, So it is attractive to classify the 1-regular graph. In the chapter 4, a complete characterization for 8-valent 1-regular Cayley Graphs whose point stabilizer is Z4 x Z2 has been presented. It is proved that there exists only 2 core-free 1-regular Cayley graphs in that case.