Weak Metacirculant,Cayley Graph And Bi-2-Metacirculant

Classification of graphs with some high level symmetric has been a hot topic in the algebraic graph theory.This dissertation focuses on weak metacirculant,Cayley graph and bi-2-metacirculant,which are organized as follows.Chapter 1 briefly introduces the history and background for weak metacirculant,Cayley graph and bi-Cayley graph,as well as our main work in this dissertation.Chapter 2 introduces some basic definitions and results regarding finite group the-ory and graph theory.In Chapter 3,we introduce absolutely split metacyclic group,and give a sufficient and necessary condition for the group being absolutely split.As a result,it is proved that a weak metacirculant of 2-power order is a metacirculant if and only if it is a split weak metacirculant.In Chapter 4,we introduce weak absolutely split metacyclic permutation groups,and prove that a graphΓis a metacirculant graph if and only if Aut(Γ)contains a weak absolutely split metacyclic transitive subgroup.Then we give a sufficient condition for a minimal split metacyclic transitive permutation group being weak absolutely split.Using this,we construct three infinite families of split weak metacirculants which are not metacirculants.For brevity,we say that a split metacirculant is a pseudo metacir-culant if it is not metacirculant.Let n be a positive integer.Our last result provides a sufficient and necessary condition for the existence of pseudo metacirculants of order n.In Chapter 5,we introduce Petersen type n-circulants.We give a characterization of G-vertex-transitive Petersen type n-circulantsΓof odd prime power order and smallest possible valency,where G≤Aut(Γ)is a metacyclic group.As a result,we construct a class of non-Cayley graphs which have a vertex-transitive non-split metacyclic group of automorphisms.In Chapter 6,a complete classification of tetravalent half-arc-transitive metacircu-lants of order a 2-power is given.In Chapter 7,a complete classification of tetravalent non-normal Cayley graphs of order 2 p~2for each prime p is given.In Chapter 8,a complete classification of connected cubic edge-transitive bi-2-metacirculants is given.In Chapter 9,we summarize the main conclusions of this dissertation and propose some open problems for further research.