Two Classes Of Groups And Their Applications In Graph Theory

Groups and graphs have always been the object of much research in mathemat-ics.But it is relatively recent to apply group theory to study graphs and apply graph theory to study groups.In this paper,we study two classes of groups and their appli-cations in graph theory.? = Cay(G,S),? is called a normal edge-transitive Cayley graph if NAt(?)(G)is transitive on the edge set of ? This concept was first proposed by Praeger,and the necessary and sufficient conditions for normal edge-transitive Cayley graphs were given.Since then,regular edge-transitive Cayley graph has attracted wide attention from scholars at home and abroad.The first work of this paper is to study the classification of tetravalent normal edge-transitive Cayley graphs on a class of 12n-order groups G ??a,6|a~4n?b~3?1,a-1ba?b-1>,and give the full automorphism group of the graph.This work generalizes the recent result of the tetravalent normal edge-transitive Cayley graphs on a class groups with the order of 6n.In 1938 R.Fruchet proved that for any given abstract group,there exists a graph with it as an full automorphism group.As an example,the second work of this paper is to prove the fact,and we proved that the full automorphism group of the Fano plane is the given non-solvable group of order 168.