Symmetric Graphs Of Valency Seven And Their Basic Normal Quotient Graphs

Let ? be a graph,and let Aut? denote its full automorphism group.A graph? is called G-arc-transitive if G ?Aut? acts transitively on arc set of ?.An arc-transitive graph is also called a symmetric graph.And ? is called a basic graph,if Aut? has no nontrivial normal subgroup N such that ? is a normal cover of the normal quotient graph ?N.There are many important properties and examples for prime-valent symmet-ric graphs,which have attracted the attention and investigation of many scholars.Especially,there are many construction methods and classifications for symmetric graphs of valency five and seven.For example,let p<q be primes,and let n be a positive integer.The symmetric graph of order p,2p,3p and pq had been clas-sified in[5,7,48,43,44],and the pentavalent symmetric graph of order 12p,2pn and 2pqn had been characterized in[23,52,15,51].Meanwhile,the study of 7-valent symmetric graphs with order 4p,4pn and 2pq can be seen in[22,40,26].In this dissertation,we classify 7-valent symmetric graphs of order 2pqn and their basic normal quotient graphs,where p<q are odd primes,and n? 2.These graphs are normal covers of an infinite class and six special symmetric graphs.In the process of proof,we also obtained the classification of two kinds of nonabelian simple groups which may has independent significance.What's more,for any given positive integer n,we also proved that there are only finitely many connected 2-arc-transitive 7-valent symmetric graphs of order 2pqn with p<q are primes and p?7,partially generalizing Theorem 1 of Conder,Li and Potocnik[8].