Semisymmetric Graphs Of Order 6p~2 And Prime Valency

The study of semisymmetric graphs has a long history, which is still a hot topic in algebraic graph theory. In this thesis, we investigate the classification problem of semisymmetric graphs of prime degree.LetΓbe a graph. The graphΓis called vertex-transitive or edge-transitive if AutΓ, the automorphism group ofΓ. acts transitively on the vertex set or the edge set. respectively. An arc ofΓis an ordered adjacent pair of vertices andΓis called arc-transitive if AutΓacts transitively on the set of arcs ofΓ. A regular graphΓthat is edge-transitive but not vertex-transitive is called a semisymmetric graph. AndΓis called half-transitive if it is vertex-transitive and edge-transitive but not arc-transitive. A regular edge-transitive graph must be arc-transitive, half-transitive or semisymmetric. Each of the above three families of edge-transitive graphs has been extensively studied in the last few decades. Thus it is meaningful to characterize and classify edge-transitive graphs.We aim to investigate semisymmetric graphs of order 6p2 and of prime degree. In [21], Lu and others gave a group-theoretic description of semisymmetric graphs of prime degree, which prove that, by considering minimal normal subgroups of the automorphism group, a semisymmetric graph of prime degree must be one of the seven types. We use this result to analyze this kind of graph which is of order 6p2 and of prime degree.In this thesis, we first give a classification of the quasiprimitive permutation group of degree dividing 3p2, and then, on the basis of the classification result, we give a complete list of semisymmetric graphs of order 6p2 and of prime degree, and prove that, for odd primes k and p, a connected graphΓof order 6p2 and degree k is semisymmetric if and only if k=3, either p=3 or p≡1(mod 6) andΓis isomorphic to one of two known graphs.In our work, we employ extensively group-theoretic results and methods. such as representation of groups, some classification results on primitive groups and on maximal subgroup of finite simple groups, and so on. We also use in the argument some combinatorial techniques and results about elementary number theory.