On The Symmetry Of Edge-transitive Cayley Graphs

The study of the symmetry of graphs is an active research field,whose main subjects are such graphs with higher symmetric properties as vertex-transitive or edge-transitive.It is of great theoretical significance to study some graphs with symmetry.An important work in determining the structure of a graph is finding its automorphism group.A graph is said to be vertex-transitive,edge-transitive or 2-arc-transitive if its automorphism group acts transitively on the vertex set,the edge set or the 2-arcs set,respectively.This thesis mainly investigates Cayley graphs with edge-transitive.And our main work of this thesis is to investigate the symmetry of the local primitive graph and the primitive 2-arc-transitive graph with square-free order.Firstly,this thesis gives a simple characterization of local primitive graph with square-free order by examining the orbit of a maximal regular subgroup of automorphism group of a graph over its vertex set.Then,this thesis structures correlation graph to meet the condition by analyzing the subgroup structures and the vertex-stabilizer subgroup of graph automorphic group,subsequently the 2-arc-transitive graphs of square-free order is classified and depicted.