Classifying One-Regular Cayley Graphs Of Valency 9

In the study of groups and graphs, the classification of transitive graphs un-der certain property has always been an active topic. The transitive property of a graph is mainly described by the action of its automorphisim group on the vertex set, edge set, arc set. Then the full automorphisim group plays a very important role in the study of its transitive property. The Cayley graph is the classical rep-resentatives for vertex-transitive graphs. In this thesis, the main research is about the classification of 1-regular Cayley graphs with valency 9.The graph Γ is called (X, l)-regular if X≤Aut(Γ) and acts regularly on its arcs, and 1-regular when X=Aut(Γ). It is easy to check that a 1-regular graph may not be 1-regular. In fact, determining whether a (X, l)-regular graph is 1-regular is very difficult.In the third Chapter of this thesis, we investigate 1-regular Cayley graphs of valency 9 with the vertex stabilizer being elementary abelian. We obtain 14 core-free (X, 1)-regular graphs of valency 9, whose vertex stabilizers are Z3 x Z3, and then determine which of them is 1-regular by constructing graphs. A complete classification for such graphs is presented. It is shown that there exists only one core-free 1-regular Cayley graphs of valency 9 with an elementary abelian vertex stabilizer.In the fourth Chapter of this thesis, we investigate 1-regular Cayley graphs of valency 9 with the vertex stabilizer Z9. We obtain 60 core-free (X, 1)-regular graphs of valency 9, whose vertex stabilizers are Z9, and then determine which of them is 1-regular basing on the order of full automorphisim group. A complete classification for such graphs is presented. It is shown that there exists 36 core-free 1-regular Cayley graphs of valency 9 with the vertex stabilizer Z9.In the end, the thesis gives a complete classification of 1-regular Cayley graphs with valency 9 up to isomorphism.