On Some Cayley Graphs With Small Valency

In the study of groups and graphs theory, the normality and regularity of a graph is always an important topic. The Cayley graph is vertex-transitive and it plays an important role in the study of groups and graphs. For a finite group G, we construct a Cayley graph with respect to its subset S, which does not contain the identity1of G and then define its normality, that is R(G)(?) Aut(Γ). In the third chapter of this thesis, we study the normality of Cayley graphs.A graph Γ is called1-regular if Aut(Γ) acts regularly on its1-arcs. In the fourth chapter and the fifth chapter of this thesis, we investigate the1-regularity of Cayley graphs with small valency.In the third chapter of this thesis, we study the normality of tetravalent con-nect Cayley graphs on the group PSL(3,2),and prove that, up to isomorphism, there exists just three tetravalent connect nonnormal Cayley graphs.In the fourth chapter of this thesis, we give a complete classification of6-valent1-regular Cayley graphs with abelian point stabilizer, and we prove that up to isomorphism, there exists only one6-valent1-regular Cayley graphs with abelian point stabilizer, which is (X, G)-(S6, S5).In the fifth chapter of this thesis, we give a classification of (X,1)-regular Cayley graphs Γ:=Cay(G, S) of valency8with Q8as its point stabilizer and also prove that if such graphs are neither normal nor bi-normal, then they must be a normal muti-cover of quotient graphs ΓN or a normal cover of5kinds of core-free Cayley graphs on G/N.