Two Classes Of Pentavalent Symmetric Graphs With Limited Order

Let Γ be a graph,G≤Aut Γ.If G is transitive on the are set of Γ,thenΓ is called G arc-transitive.In particularly,if G = Aut Γ,then Γ is called arc-transitive or symmetric.The classification of symmetric graphs with some limited order has always been one of the central problems in algebraic graph theory.For example,[1,2,3,4,5]have determined the classification of symmetric graphs with order p,2p,3p,pq respectively.In the study of classification of symmetric graphs,if there has no limited in the valency of the graphs,the work will be difficult.So the classification problem in syinmetric graphs with small valency and order-limited becomes the core problem in this field,especially the symmetric graphs of valency 5.For example,in 2011,X.H Hua et al.have determined the classification of pentavalent symmetric graphs of order 2pq.In 2016,Y.Q Feng et al.have determined the classification of pentavalent symmetric graphs of order 2pn.Recently,H.L Liu et al.have determined the pentavalent symmetric graphs with order 2p2q,where p,q are distinct odd primes.On the basis of above works,the first work of this thesis is to determine the classification of pentavalent symmetric graphs of order 2p2q2,where p,q are distinct odd primes.By discussing the group action on graphs and applying the knowledge of permutation groups and algebraic graph theory and Magma software,we proved that the graphs in this case can only be Cayley graphs on 6 generalized Dihedral groups with order 2p2q2 and arc-regular graphs.Furthermore,we give the full automorphism groups and point stabilizers of these graphs,and construct examples.In 2011,X.H Hua et al.have determined the pentavalent symmetric graphs of order 8p in[6].In 2016,J.M.Pan et al.has determined the pentavalent symmetric graphs of order 4p2 in[12].On the basis of above works,the second work of this thesis is to determine the classification of pentavalent symmetric graphs of order 8p2,where p is a prime.The research shows that,the pentavalent symmetric graphs of order 8p2 exists if and only if p = 2.,3.Furthermore,only two pent,avalent symmetric graphs exist with order 32 when p = 2 up to isomorphism.Only two pentava.lent symmetric graphs exist with order 72 when p = 3 up to isomorphism.