Combinatorial Designs With Good Transitivity And Semisymmetric Graphs

The research of finite permutation group and combinatorial structure is an active and interdisciplinary subject which blends algebra and combinatorics in recent decades.The study of designs with good transitivity is still an active subject in this interdisciplinary subject,which has great meaning in theory and in practice.In algebraic graph theory,the classification and construction of semisymmetric graphs is a very important and active research topic with important theoretical significance.In this thesis,on the one hand,we use the properties of the finite shomogeneous permutation groups to study designs with s-flag-transitivity and block-transitivity.On the other hand,we use the good transitivity of designs to construct semisymmetric graphs.In particular,we construct several worthy semisymmetric graphs by using the s-flag-transitivity of some designs.The specific content of this thesis is as follows:First of all,the research background and current development of the combinatorial design and permutation groups and the semisymmetric graphs are summarized.Also,the related basic theories and some classical conclusions of groups,graphs and combinatorial designs are presented.Moreover,we also fix some notations which are used frequently in this dissertation.Then,we consider the problem about the classification of designs with sflag-transitivity.On the one hand,we completely classify the s-flag-transitive Steiner designs and s-flag-transitive t-designs with s=4 and 5 based on the classification results of finite s-homogeneous permutation groups.On the other hand,based on the fact that the incidence graph of symmetric design is a balanced bipartite graph,we define the s-symmetric design as the design whose s-incidence graph is a balanced bipartite graph.And we also give a characterization on the s-flag-transitive s-symmetric designs with s>2.Next,we consider using the s-flag-transitivity of designs to construct semisymmetric graphs.First of all,we study the property of the s-incidence graph,denoted by IGs(D),of design D,and we conclude that the s-incidence graph of an s-flag-transitive design is edge-transitive.Next,we study the symmetry of IGs(D).For s=1,we prove that IG1(D)is a semisymmetric graph if and only if D is a non-self-dual symmetric design;for a non-trivial s-flagtransitive s-symmetric design,we also prove that IGs(D)is a semisymmetric graph or a balanced bipartite symmetric graph;and for s>1,we give a sufficient and necessary condition for the s-incidence graphs of Steiner designs to be semisymmetric graphs.Then we construct several worthy semisymmetric graphs by using the characterization results of s-flag-transitive s-symmetric designs.Finally,in order to construct more semisymmetric graphs from designs,we define the generalized s-incidence graphs of designs,denoted by GIGS(D),and we find a series of examples of semisymmetric graphs from GIGs(D)by using the classification results of s-flag-transitive Steiner designs.Finally,we study the problem about block-transitive Steiner design.The famous Camina-Gagen theorem of designs theory is that if G acts as a blocktransitive group of automorphisms of a 2-(v,k,1)design with k dividing v,then G is flag-transitive.We generalize this theorem to other Steiner designs,for 3<t ≤6,we prove that if G acts as a block-transitive group of automorphisms of a Sterner t-design with the block size k dividing the pointnumber v,then G acting on this Sterner design is also flag-transitive.Besides,for a block-transitive Sterner 3-design with the point-number v being odd or v ≡ 0(mod 4),we prove that if the block size k divides v,then this design also is flag-transitive.