Heptavaient Symmetric Graphs Admitting Vertex-transitive Non-abelian Simple Groups

A graph ? is said to be symmetric(or arc-transitive)if its au-tomorphism group Aut(?)acts transitively on the are set of ?.The symmetry of graphs has been being a hot issue problem in studying groups and graphs,which mainly depends on some transitive properties of the automorphism groups of the graphs to describe.With the classification of the finite simple groups,Cayley graphs and coset graphs of simple groups have been thoroughly studied.In this paper,we consider the heptavalent symmetric graphs.Let G be a,finite non-abelian simple group and let ? be a connected heptavalent symmetric G-vertex-transitive graphs with solvable stabilizers.In this paper,we prove that either G(?)Aut(?)or Aut(?)contains a non-abelian simple normal subgroup T such that G ? T and(G,T)=(An-1,An)with n=7,22·3,2 · 7,2 · 32,3 ·7,22 · 7,22 ·32,2 · 3 · 7,32 · 7,22 · 3 · 7,2 · 32 · 7 or 22 · 32 · 7.Chapter 1,we will introduce some basic definitions of finite group theory and graph theory,and the relative background of symmetric graphs of small valency.Chapter 2,we will describe the structure of the solvable vertex stabilizers of connccted heptavalent symmetric graphs.and introduce the definitions of the quotient graph,non-abelian simple groups and the covering groups.Chapter 3,we will prove the main results of this paper.In this chapter,we obtain the pairs of non-abelian simple groups(G,T)such that 22 · 32 · 7 can be divided by the index of G in T and then finished the proof of the main theorem.