Vertex Transitive Locally Primitive Graphs Of Order Twice A Prime Square

Let ? be a,graph,G ?<Aut?.The graph ? is called G-locally primitive if the stabilizer subgroup G? of G acts primitively on the neighbor set of ? for arbitrary vertex ??V ?.Especially,? is called locally primitive if G = Aut ?.In this thesis,vertex transitive locally primitive graphs of order twice a prime square is investigated.By definition,a 2-arc transitive graph must be locally primitive arc-transitive,which is one of the central questions in algebraic graph theory.Trivalent vertex transitive graph of order twice a prime square has been studied by Jinxin Zhou in 2008.Pentavalent symmetric graphs of order 2pq have been classified by Xiaohui Hua in 2011.Pentavalent symmetric graphs of order 2p2 have been investigated by Jiangmin Pan.etc in 2015.Pentavalent symmetric graphs of order 2pn have been investigated by Yanquan Feng.etc in 2016.Obviously,prime valency symmetric graph must be locally primitive symmetric graph.Under the background as above,this thesis studied locally primitive symmetric graphs of order 2p2,and generalized the partial results mentioned above.A typical method studying locally primitive vertex graph is put forward by Praeger in 1992,and developed into Global-action analysis.In this thesis,locally primitive symmetry graph of order 2p2 is studied by dividing into three steps,that is vertex quasiprimitive,vertex bi-quasiprimitive and general case.In research process,we applied the permutation group theory and the cover of the quotient graph,and obtained a complete classification about locally primitive symmetric graph of order 2p2,and observed that all of locally primitive symmetric graphs of order 2p2 are Cayley graph except for HS(50).