Several Classes Of Transitive Graphs And Their Relevant Arc-transitive Covers

This dissertation is devoted to study two important and relative topics in algebraic graph theory:the transitive graphs and their relevant arc-transitive covers.Characterizing the cover of a graph is an important method in algebraic graph theory,and the research essence is based on the theory of group extension.However,determining group extension is always a difficult problem in the field of finite group.Therefore,characterizing or classifying covers of arc-transitive graphs is one of the difficulties in algebraic graph theory.The first work in this dissertation is to study covers of several arc-transitive graphs by using the theory of group extension.In particular,we characterize the arc-transitive Schur multiplier covers and metacyclic covers of complete graph,and the primitive covers of arc-transitive graphs which admit a group with two direct-product factors as an arc-transitive automorphism group.If the full automorphism group of a graph contains a subgroup which is regular on its vertices,then the graph is a Cayley graph on this regular subgroup.Based on the symmetry and simplicity of Cayley graphs,it has always been a hot topic in algebraic graph theory.It is well-known that a Cayley graph is vertex-transitive,but not necessarily arc-transitive.Thus the second work in this dissertation is to study several families of arc-transitive Cayley graphs.We classify the cubic arc-transitive Cayley graphs on Frobenius group G?Zpd:Zn with p a prime and d,n integers,and further characterize pentavalent and heptavalent arc-transitive Cayley graphs on G for which the vertex stabilizer G1 is soluble.A graph is said to be half-arc-transitive if its automorphism group is transitive on its vertex set and edge set but not on arc set.In the field of algebraic graph,it is always an enduring topic to construct half-arc-transitive graph.In fact,4 is the smallest admissible valency for a half-arc-transitive graph.The tetravalent half-arc-transitive graphs of order pq and p2q have been classified,where p and q are odd primes.The third work of the dissertation is to characterize tetravalent half-arc-transitive graph of order p2q2,which generalizes the above results.It is known to all,every symmetric graph can be represented as a coset graph related to its automorphism group.On the other hand,each finite group may be as an automorphism group for many symmetric graphs.Therefore,it is a meaningful work to study symmetric graphs of small valency with certain orders.In recent decades,this problem has been widely studied.For example,cubic,pentavalent and heptavalent arc-transitive graphs of square-free order,and cubic arc-transitive graphs of cube-free order have been classified.Afterwards,pentavalent arc-transitive graphs of 2pq and 4pq have been classified,where p and q are odd primes.Under the above background,we classify the pentavalent arc-transitive graphs of order 2p2q.Recently,small valent arc-transitive graphs of order 4p and 4p2 have been classified.Thus the fifth work of this dissertation is to generalize the results to prime-valent arc-transitive graphs.We classify prime-valent arc-transitive basic graphs with order 4p and 4p~2.