Normal Edge-Transitive Cayley Graphs On Nonabelian Groups Of Order 2p~2

Group theory is one of the greatest concepts in mathematical theory,and graph theory which as an ancient branch of mathematics has penetrated into many fields of modern science.As two very important mathematical research objects,there exists a very close relationship between groups and graphs.The study of groups by graphs and the study of graphs by groups originated in the beginning of last century,German graph theorist R.Frucht obtained a pioneering result,for any abstract group,there exists a graph whose automorphism group is the given group,which started the prelude to combine the groups and graphs.In recent years,graphs have been applied in the study of permutation groups,and obtained many excellent results,which greatly promoted the development of algebraic graph theory.In algebraic graph theory,Cayley graphs are important research objects.They are successful examples of studying graphs by group theory.At present,one of the hot issues in the study of Cayley graphs is the study of its normality,especially the study of normal Cayley graphs and normal edge transitive Cayley graphs.The concept of normal Cayley graphs was first introduced by Chinese scholar Xu Mingyao in the 1990s,and the concept of normal edge transitive Cayley graphs was firstly proposed by Praeger in 1999,and the necessary and sufficient conditions of normal edge transitive Cayley graphs were given.Since then,the study of normal Cayley graphs and normal edge transitive Cayley graphs has been getting attention by many scholars in Chinese and abroad.In this thesis,we mainly study the normal edge transitive Cayley graphs on nonabelian groups of order 2p~2.In 2021,Zhou Jinxin et al described and classified tetravalent non-narmal Cayley graphs of order 2p~2,so the structures of nonabelian groups of order 2p~2 were clear.In this thesis,we investigated the trivalent and tetravalent normal edge-transitive Cayley graphs on nonabelian groups of order 2p~2 firstly,and finally the normal edge transitive Cayley graphs on these groups are described and classified.