The Normality Of Some Small Valent Cayley Graphs

In the study of groups and graphs theory, Cayley graphs always play an important role. Given a finite group G, a basic problem is to determine all normal or nonnormal Cayley (di)graphs of G. In some studies, we often need to determine the full auto-morphism groups of Cayley graphs. Normal Cayley graphs play an important role in determining their full automorphism groups. Normal Cayley graphs have the smallest possible full automorphism groups, that is Aut(r)=R(G)Aut(G, S). Hence, the full automorphism group of a normal Cayley graph Cay(G, S) can only be determined by the group G and subset S. Meanwhile, the concept of normality of Cayley digraphs is known to be important for the study of arc-transitive graphs and half-transitive graphs. Determining the normality of Cayley graph is one of difficult problems in graph theory in general. The content of this thesis goes around the normality of Cayley graphs.First, we study the normality of cubic Cayley graphs of the dihedral group D2p2q2, where both p>q are odd primes. We prove that those Cayley graphs are normal. As an application, we also prove that their full automorphism groups are solvable.Second, we study the normality of Cayley digraphs of valency2of finite groups of order10p, where p is prime. We prove that all but the following cases are normal.(1)G=Z10p=<a>, S={α,α5p+1};(2) G=Z5p×Z2=<a>×(u),S={α, αu};(3) G=<α,6, c|a5=b5=c2=1,[a, b]=[a,c]=1,c-1bc=b-1>, S={cα, cbα-1}.Third, we study the normality of tetravalent Cayley graphs of finite groups of order2p2, where p is prime. Assume that F=Cay(G, S) is not arc-transitive and is a2-group. Then(1) p=5, r is normal, in particular, Aut(r) is solvable;(2) p=5, Aut(r) has a normal subgroup N isomorphic to Zp, r is a cover rN, FN=K5,5-5K1.Finally, the normality of cubic Cayley graphs of the dihedral group D20P is studied.