The M-Competition Indices Of Some Special Doubly Symmetric Primitive Graphs

Combinatorial matrix theory is a new branch of mathematics which has been developed in the last 40 years.Its core content is to study the combinatorial properties of matrices.Index theory is one of the most important aspects of combinatorial matrix theory.It not only has close relationship to many branches of mathematics such as matrix theory,graph theory,number theory,probability theory,dynamic system,etc.,but also has remarkable applications in practice,for example,finite automata design,memory less communication systems,channel assignment,and modeling of complex economic and energy systems.The m-competition index of a primitive digraph(a primitive matrix)was introduced by H.K.Kim in 2010 to study the properties of the k-step competition graph of a directed graph.It is a generalization of the scrambling index and the primitive exponent for a primitive digraph.In this thesis,we study the m-competition indices of some special doubly symmetric primitive graphs.This thesis is structured as follows:In chapter 1,we first introduce the background and the significance of the combination matrix index theory,and then describe the current status of research of the m-competition index including the scrambling index(i.e.1-competition indices).We also sketch out the briefly content of this thesis.In chapter 2,by using the relationship between the scrambling index and the exponent for a symmetric primitive digraph,we determine the scrambling indices(i.e.1-competition indices)of some doubly symmetric primitive graphs.In chapter 3,the m-competition indices of two doubly symmetric primitive simple graphs and a doubly symmetric primitive graph with loops are determined.In chapter 4,we summarize the main results of this thesis,and discuss a perspective of further research work.The graphs studied in this thesis have played an important role in the study of the generalized exponents of doubly symmetric primitive matrices.We hope that the results of this thesis will contribute to the further study of the m-competition index of doubly symmetric primitive graphs.