On The Existence Of λ-configurations

λ-configuration is a kind of finite incidence structure in combinatorial designs. It is connected with many combinatorial structure closely. The construction and existence ofλ-configurations could be applied to the key pre-distribution schemes in wireless sensor networks.In this thesis, two construction methods ofλ-configurations are introduced:differ-ence triangle method and group divisible design method. We investigate the existence of some kinds ofλ-configurations by using the two methods mainly.There are four chapters in the thesis.In Chapter 1, the relevant definitions and theorems aboutλ-configurations are intro-duced firstly. Secondly, we introduce the basic principle of operation about the wireless sensor networks, and the application ofλ-configuration in key pre-distribution schemes. At last, some early results and new progress about the spatial configuration are pre-sented.In Chapter 2, the difference triangle construction method is introduced in Section 1. The existence of some kinds ofλ-configurations are investigated by using the methods in the following sections. In Section 2, some new symmetric spatial configurations are constructed and the existence table of symmetric spatial configurations is updated. In Sections 3 and 4, the existence of symmetric 3-configurations and 4-configurations are investigated and two existence tables are obtained.In Chapter 3,λ-configurations are constructed from uniform group divisible de-signs. Some basic definitions are introduced in Section 1 and two construction meth-ods are given in Section 2. The basic method is establishing the contacts betweenλ-configurations and uniform group divisible designs. The existence problems of someλ-configurations could be solved by using the method.In Chapter 4, the main results of the thesis are summarized, and the further research problems are presented.