The Investigation For One-Factor Block Graph With The Action Of Simple Group

The symmetry of graphs has been being a very hot issue problem in studying groups and graphs and it mainly depends on some transitive properties of acting by the automorphism groups of the graphs to describe. The Cayley graph and the Sabidussi coset graph are two classical representatives for these graphs.For Cayley graph, a very interesting situation is that its automorphism group is imprimitive as it acts on V(X) when the nonsymmetric cubic graphs of goup G is not GRR, so we can get two different kinds of block graph of the graph: 1- factor block graph and another basic cycle block graph. We will find some interesting symmetry properties for the two kinds of block graph by studying. For example, the one- regular property and high arc- transitive property, and so on. In fact, the two different kinds of block graph are all Sabidussi coset graphs of group G. Furthermore, The material symmetry studying of Cayley graphs of small valencies for some groups takes a very important place in studying groups and graphs.Those questions are the main purpose of my thesis, we get some elementary properties of one- factor block graph and we also give a complete classification and some elementary properties of the Cayley graphs of small valencies of some groups.we have the following theorems as the main results of this thesis.1. We study the construction of 1- factor block graph with the action of simple group G, especially the 1- regular and noncayley graph properities;2. We construct the infinite families of 1- transitive with valencies 3 by finite nonabelian simple groups; 3. We give a complete classification of all arc- transitive Cayley graphs on A₆ of valencies 3 and 4 and the description of some symmetric properties;4. We give some examples of GRR of A₆;5. We confirm the arc- transitive Cayley graphs on S₄ of valencies 3 and the description of their properties.The method used in this thesis is mainly group- theoretic. For concepts of group theory and algebraic graph theory we refer readers to [1, 2, 3]...