Classification Of 2-arc-transitive(Bipartite) Graphs

In the thesis,we investigate the classification problem of 2-arc-transitive(bipartite)graphs.Let Γ be a graph.The vertex set,edge set and full automorphism group of Γ will be denoted by V(Γ),E(Γ)and Aut(Γ),respectively.An arc of Γ is an ordered pair of adjacent vertices,and a 2-arc is a triple(α,β,γ)of vertices with {α,β},{β,γ} E E(Γ)and α≠γ.The graph is called(G,2)-arc-transitive if G≤Aut(Γ)is transitive on the set of 2-arcs of Γ.A graph Γ of valency at least 3 is called a basic 2-arctransitive graph if it has 2-arc-transitive group G such that every minimal normal subgroup of G has at most two orbits on V(Γ).Given a group H,take a subset S of H.The bi-Cayley graph over H with respect to S,denoted by BiCay(H,S),is defined as a graph whose vertex set is the union of following two copies of H:H0={h0 | h ∈H｝,H1=｛h1| h∈ H},and whose edge set is {{h0,(sh)1} |h0∈H0,h1∈H1,s ∈ S}.W.T.Tutte gave a beautiful result in 1947:cubic graphs are at most 5-arctransitive,and this result is considered to be the source of the finite s-arc transitive study.Since then,the classification of 2-arc transitive graphs has attracted widespread attention and has gradually become a very active research topic in algebraic graph theory.C.E.Praeger[45,46]observed that a connected 2-arc-transitive graph is a cover of some basic 2-arc-transitive graphs,and proposed the problem of classifying all finite basic 2-arc-transitive graphs.Let Γ be a basic(G,2)-arc-transitive graph.For {α,β｝∈ E(Γ).If Γ is not bipartite,then G is a quasiprimitive group on V(Γ).For this case,C.E.Praeger[45]proved that G is one of the HA,AS,PA and TW.Now assume that Γ is bipartite,then G has a subgroup G+ of index 2 with two orbits on V(Γ),say V1 and V2,the two parts of the bipartition of V(Γ).C.E.Praeger[46]proved that either Γ is a complete bipartite graph,or G+is faithful on both parts of Γ.In the thesis,we focus on research the bi-primitive 2-arc-transitive bi-Cayley graphs and basic 2-arc-transitive graphs of order ras or 2ras.There have been classification results for primitive or bi-primitive s-arc-transitive graphs in the literature.One of the remarkable achievements is that Li gave a complete classification of primitive and bi-primitive s-arc-transitive graphs for s≥ 4 in[23].In 2010,Li et al.completely classified primitive and bi-primitive graphs s-arcregular graphs in[30].Li and Zhang[32]gave the classification of primitive and bi-primitive graphs Γ such that Aut(Γ)acts transitively on the 2-paths but not on the 2-arcs of Γ in 2012.Recently,all primitive s-arc-transitive non-normal Cayley graphs are classified in[39].In this paper,we mainly investigate the bi-primitive 2-arc-transitive bi-Cayley graphs.First of all,the action of Aut(Γ)on the V(Γ)biprimitive is reduced to the fact that Aut(Γ)is almost simple.Next,according to the classification of almost simple primitive permutation groups with a regular subgroup[41],combined with the knowledge of related group theory,especially permutation group theory and graph theory,a complete classification of bi-primitive non-normal 2-arc-transitive bi-Cayley graphs is given.In the second part of this paper,we focus on the basic 2-arc-transitive graphs of order ras or 2ras,where r and s are distinct primes.Let Γ be a basic(G,2)-arctransitive graph,where either |V(Γ)|=ras or |V(Γ)|=2ras(if Γ is bipartite).If r=s then Γ is classified in[21].Thus we consider the case where r≠s.Assume that G is almost simple.Next,according to the classification of permutation groups of degree a product of two prime-powers,we present a complete classification for basic 2-arctransitive non-bipartite graphs of order ras and basic 2-arc-transitive bipartite graphs of order 2ras.