Cayley Graphs Of Small Valencies Of Some Solvable Groups

The symmetry of graphs has been being a very hot issue problem in studying groups and graphs and Cayley graph plays an important role in the reseach work. The purpose of this thesis is studying the symmetry properties of Cayley graph of some solvable groups,such as metacyclic groups with order 3p,finite 2-groups,the groups with order 2ⁿp,the groups with odd order and metacyclic groups with order qp.At first,This thesis determines the full automorphism groups of connected Cayley graphs of valency 4 of metacyclic groups with order 3p(p＞3 is prime)and then obtain their CI property,normality and arc-transitive property.We prove:For the vertex-stabilizers A₁ of the full automorphism groups of metacyclic groups with order 3p,there is |A₁|≤2 except for a graph,which is isomorphic to a coset graph of the special projective group PSL(2,7)with respect to one of its Sylow 2-subgroups. As the corollaries of this conclusion,we obtain that all metacyclic groups with order 3p are weak 4-CI groups and the connected Cayley graphs of valency 4 of metacyclic groups with order 3p are normal but no arc-transitive except for the above graph.Secondly,For the connected Cayley digraphs of valency 2 of finite 2-groups, This thesis proves that they are all normal;In addition,This thesis gives a sufficient condition that the connected Cayley digraphs of valency 2 of groups with 2ⁿp and odd order are normal. At last,For the connected edge-transitive Cayley digraphs of non-abelian groups with order qp,This thesis proves that there exist directed Hamilton circles.The method used in this thesis is mainly group-theoretic.For concepts of group theory and algebraic graph theory we refer the reader to[1,2,3].