The Qualities Of Small Valency Cayley Graphs On Sub-simple Groups And Character Simple Groups

The symmetry of graphs has always been a hot issue in studying group and graph. It is mainly described by some transitivity possessed by the automorphism groups of the graphs. The classical representatives of these graphs are Cayley graph and Sabidussi coset graph. Since simple groups are the base to consititute finite group, and they have the inner prosperity of group, studying the simple group has been a very hot issue problem. In [1, 2], Caiheng Li and Shangjin Xu proved there are only two nonnormal connected cubic arc-transitive Cayley graphs of nonabelian simple group, which two graphs are 5-arc-transitive and their order are (47!)/2. These results not only determine the normality of connected cubic arc-transitive Cayley graph on nonabelian simple group, but also give the compete classical of connected cubic arc-transitive Cayley graph of nonabelian simple group. Nonsolvable sub-simples group is a nonsolvable finite group with only a normal subgroup, and character simple group is a product of isomorphic simple groups, both of them have close relation with simple group. So, it is very natural and interesting to research their small valency Cayley graphs.The symmetry of Cayley graphs depends on the information how deeply we know from their full automorphism groups. As we know, it is very difficult to determine the automorphism group of graphs, and it is a fundamental problem. Till now, only a few of kinds of group with some characters are known. In additon, the normality is also a fundamental problem for the symmetry of Cayley graphs. In this thesis, about these porblems, we study the autmorphism groups and normality of small valency Cayley graphs on 2-dimensional linear groups which they are classical representatives of nonsolvable sub-simple groups.Investigating fnite CI-graphs has been a currently very active topic in the area of Cayley graphs since Adam put forward a conjecture which each cyclic group is a CI-group in 1967. Caiheng Li proved that all simple groups are 3-CI groups, accordingly, we want to know whether nonsolvable sub-simple groups are 3-CI groups? About this problem, we prove S₅ which its order is the smallest in nonsovable sub-simple groups is not 3-CI group.So, in this thesis, my main results are the following items:Questionâ… Research the structure of automorphism groups and normality of connected cubic and tetravelent edge-transitive Cayley graphs on PGL(2,p)(p is a prime). Questionâ…¡Determine the normality and CI-property of connected cubic Cayley graphs on S₅.Questionâ…¢Discuss the normality of connected cubic arc-transitive Cayley graphs of nonsolvable character simple groups, and get the Cayley graphs on character simple groups are nonnormal only when G≤S₂₄ and their cubic arc-transitive Caylye graphs are 5-arc transitive graphs.The method used in this thesis is mainly group-theoretic. For concepts of group theory and algebraic graph theory we refer the readers to [3, 4, 5].