| Boolean matrix is a matrix composed of only 0 and 1.Boolean matrix is widely used in many fields,such as logic,communication engineering,combinatorial theory,graph theory,semigroup theory,fuzzy set theory and computer science.Boolean network is a transient dynamic systemf f＝（f1,…,fn）:F2n→F2n.Boolean network also plays a key role in computer science,psychology,circuit design and other fields.The main research contents of this paper are:graph theoretic properties of idempotent Boolean matrices and square roots of Boolean matrices,graph theoretic properties of nilpotent Boolean matrices and its permanent,Normal AND-NOT fixed point properties of Boolean networks.The details are as follows:In the first chapter,we mainly introduce research background and significance of Boolean matrices and Boolean network.In chapter 2,we give some relevant basic knowledge,definitions and symbols.In chapter 3,we mainly introduce some properties of idempotent Boolean matrices and square roots of Boolean matrices.This chapter mainly gives the characterization of idempotent Boolean matrix from the view of graph theory,and gives the method of construct a new square root by using the square root of Boolean matrix.In addition,we also study the idempotence of intersection and union of maximal transitive relation matrices.In chapter 4,we give the sufficient and necessary conditions of the permanent of Boolean matrices be 1 from the view of graph theory.According to the graph structure of nilpotent Boolean matrices,the directed graphs corresponding to the adjoint matrices are obtained,and thus the adjoint matrices are constructed,and we describe some properties of the adjoint matrices of nilpotent Boolean matrices.In chapter 5,we study the normal AND-NOT Boolean network.This paper introduces the support matrix and exponential matrix of normal AND-NOT Boolean network,and the relationship between these two matrices.Moreover,we study the structure of the state space graph of normal AND-NOT Boolean network. |
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