Prime Valent Edge-transitive Graphs Of Order Four Times A Square-free Integer

LetΓbe a graph.Use V（Γ）,E（Γ）,Arc（Γ）and Aut（Γ）to denote the vertex set,edge set,arc set and the full automotphism group ofΓ.Take G≤Aut（Γ）.If G is transitive on V（Γ）,E（Γ）or Arc（Γ）,then we say that,Γis G-vertex-transitive,G-edge-transitive,or G-arc-transitive,respectively.If G is regular on E（Γ）or Arc（Γ）,we say thatΓis G-edge-regular or G-arc-regular,respectively.If aΓwith regular valency is G-edge-transitive but not G-vertex-transitive,thenΓis said to be G-semisymmetric.In the case when G=Aut（Γ）,an G-vertex-transitive graph,G-edge-transitive graph,G-arc-transitive graph,G-edge-regular graph,G-arc-regular graph or G-semisymmetric graph will be simply called a vertex-transitive graph,edge-transitive graph,arc-transitive graph,edge-regular graph,arc-regular graph or semisymmetric graph,respectively.Edge-transitive graphs with given order and valency are an active topic in the re-search field of group and graph.In[G.X.Liu,Z.P.Lu.On edge-transitive cubic graphs of square-free order[J].European Journal of Combinatorics,2015,45:41-46]^[2],a classifi-cation of cubic edge-transitive graphs of square-free order was given,and in[J.M.Pan,Y.Liu.There exist no arc-regular prime-valent graphs of order four times an odd square-free integer[J].Discrete Mathematics,2013,313:2575–2581]^[1],it was prove that there do not exist arc-regular prime-valent graphs of order four times an odd square-free integer.Motivated by these works,this paper is study of prime valent of edge-transitive graph-s of order four times an odd square-free integer.We first prove that there do not exist edge-regular prime-valent graphs of order four times an odd square-free integer,and then we determine the automorphism groups of cubic symmetric graphs of order four times an odd square-free integer.A new infinite family of cubic symmetric graphs are constructed.This paper is organized as follows:Chapter 1 introduces the research background and some basic problems,and states our main results.Chapter 2 gives some related concepts and preliminary results on graph theory and group theory.Chapter 3 proves that there do not exist edge-regular prime-valent graphs of order four times an odd square-free integer.Chapter 4 determines all automorphism groups of cubic semi-symmetric graph graphs of order four times an odd square-free integer.Chapter 5 poses some possible problems for further study.