Several Families Of Edge-transitive Graphs

The thesis investigates the application of group theory on graph theory, whose subjects are graphs with some symmetric properties, and whose main method is applying automorphism group of a graph to analyze its symmetries. The main results are classifications and enumerations of several families of edge-transitive graphs.Chapter 1 introduces some basic definitions of group theory and graph theory, main results and relative backgrounds of the research.Chapter 2 classifies cubic semisymmetric graphs of order 8p3 and gives some properties of cubic edge transitive bipartite graphs of order 2qpr(q< p,r=2,3). A graph is called semisymmetric if its automorphism group is transitive on its edge set but not transitive on vertex set. By Folkman [J. Combin. Theory 3 (1967) 215-232], there is no semisymmetric graph of order 2p or 2p2 for a prime p, and by Malnic et al. [Discrete Math.274 (2004) 187-198], there exists a unique cubic semisymmetric graph of order 2p3, the so called Gray graph of order 54. In this chapter it is shown that there is no connected cubic semisymmetric graph of order 4p3 and there exists a unique cubic semisymmetric graph of order 8p3, which is a Z2×Z2-covering of the Gray graph. For a cubic edge-transitve bipartite graph of order 2qpr, we not only prove that its automorphism group has a normal Sylow p-subgroup and has order no more than 6qpr, but also prove that it is a normal bi-Cayley graph on a group G of order qpT. And using this, we get some sufficient and necessary conditons for existence of cubic semisymmetric graph of order 2qp2.Chater 3 classifies connected pentavalent symmetric graphs of order 2pq and 8p, where p, q are distinct primes. A graph is symmetric if its automorphism group acts transitively on its arcs of the graph. For pentavalent symmetric graphs of order 2pq, there are two connected pentavalent symmetric graphs of order 4p, that is K6,6—6K2 or Icohedral I, and for odd primes p and q, there is an infinite family of connected pentavalent symmetric graphs of order 2pq with solvable automorphism groups and there are seven sporadic ones with nonsolvable automorphism groups. In particular, four of the seven sporadic ones are 2-transitive and bi-primitive. For pentavalent symmetric graphs of order 8p, it is Clcbsh graph CL16, a standard double cover of Icohedron, or the coset graph of G=PSL2(31) relative to a subgroup A5, denoted by C248.Chapter 4 investigates isomorphisms and automorphisms of the vertex prim-itive and bi-primitive 2-arc transitive graphs admitting a 2-dimenensional linear group. A graph is said to be primitive if its automorphism group is primitive on the vertex set, and bi-primitive if it is a bipartite graph, and its automorphism group which fixes each part is primitive on each part. We not only classify the vertex primitive and bi-primitive 2-arc transitive graphs admitting a 2-dimensional linear group but also give the number of non-isomorphism graphs and the automorphism group of these graphs.