On The Graph Structure Of Several Kinds Of Rings And Semi-rings

Let X is a finite set,f is a function of X,and finite dynamical system?X,f?is widely used in cellular automaton,computational science,computational biology,cybernetics and computer simulation.In this paper,we study the correlation graph structure of linear dynamical system and Boolean network.In this thesis,we study the iterated graph of two-dimensional linear dynamical system over the ring of integers modulo the power of a prime,as well as the zero-divisor graph of the quotient ring of an imaginary quadratic integer ring with class number one modulo the power of a maximal ideal,and the zero-divisor graph of then?n Boolean matrix semi-ring.The structures of these graphs are completely characterized.The following is the main content of this thesis.The first chapter introduces the research background and the symbols involved in the article and some basic knowledge.The second chapter investigates the two-dimensional linear dynamical system a?28??a₁,a₂??Z²_pt.In order to facilitate the visualization of this system,we associate a graph?38??A?on it,whose vertices are all elements ofZ_pt,and two distinct verticesaandbare joint by a directed edge whenever?A?28??.In this chapter,we state the criterion for vertices with the in-degree greater than 0 in?38??A?and give the formula of in-degree.Moreover,we completely describe the structure of circles of?38??A?when A is an idempotent matrix,or a nilpotent matrix,or a diagonal matrix.Because the extended algebraic integer ring of rational number field is an important research object of dynamical system,In the third chapter,we study the zero-divisor graph structure of the quotient ringO_K/xⁿof the algebraic integer ringO_K of the quadratic extension K?28?Q?d??d?27?0?of the real number field Q with the class number 1,where?is a prime element in K.It is proved that the diameter of the zero-divisor graph of this kind of quotient ring is no more than 2,and the planarity of the zero-divisor graph is studied.Boolean matrix is a common research object in Boolean network.In the fourth chapter,we present a complete description of the undirected zero-divisor graph,the complement graph of the undirected zero-divisor graph,as well as the directed zero-divisor graph of then?n matrices over the binary Boolean semiring B?28?{0,1}.The diameter,girth,planarity and degree of special vertices of these graphs are determined.