Finite Edge-transitive Metacirculants, And Locally-quasiprimitive Graphs

Classifying the finite graphs has been a hot topic in the algebraic graph theory.This thesis focus on the finite metacirculants, and the main tools include the theory ofsubgraphs and quotient graphs, permutation groups and abstract groups, the classifica-tion of simple groups, and the theory of the factors of groups. At the same time, thelocally-quasiprimitive graphs are studied.The concept of metacirculant was introduced by Alspach and Parsons (1982)in[5].A graph Γ=(V, E) is called an (m, n)-metacirculant, where m, n are positiveintegers, if Γ is of order|V|=mn and has two automorphisms ρ, σ such that: ρ issemiregular and has n orbits on V; σ cyclically permutes the n orbits of ρ and nor-malizes ρ; σnfixes at least one vertex of Γ. About the finite edge-transitive metacir-culants,the main results are listed in the following:1. The study on general metacirculants. The definition given by Alspach and Par-sons is complex and not convenient for the study of the metacirculants. In thisthesis, a criteria in group theory is given: A graph is a metacirculant if and onlyif the automorphism of this graph contains a transitive split metacyclic subgroup.This provides a simple definition for the metacirculant.2. Characterize few more important classes of metacirculants. Especially, given theclassification of the metacirculants of Frobenius metacyclic groups. For example,we proved that there are exactlyΦ(m)2edge-transitive Cayley graphs with valency2k (k p1) of Frobenius group Zpd:Zm, where Φ(m) is the Euler Φ function.This covers many results given before.3. Characterize the bi-quasiprimitive permutation groups which contain regular di-hedral subgroups. This provides a powerful tool for the study of dihedrants. Asan application, we characterize the edge-transitive dihedrants of order2pdwithvalency coprime to p.4. In this thesis, a few infinite classes of half-transitive graphs are constructed. For example, it is proved that a few important edge-transitive metacirculantsof Frobenius metacyclic groups are almost half-transitive. Inspecting by thisbreakthrough, we conjectured that, except for a few known special graphs, alledge-transitive Frobeniuns metacirculants of odd order are half-transitive.A graph is called G-locally-quasiprimitive, if GΓ(α)αis quasiprimitve. In this the-sis the global action and the structural information of such groups G is studied, ex-tending the previous results for locally-primitive graphs and vertex-transitive locally-quasiprimitive graphs.