Singularity And Eigenvectors Of Mixed Graphs

In 70's of the 20th century, Fiedler defines the second least Laplacian eigenvalue of a graph as its"algebra connectivity", which is considered as a quantitative measurement of graph connectivity. The eigenvectors corresponding to the algebra connectivity is called Fiedler vectors, and has a very beautiful combinatorial property given by Fiedler. From then on, the domestic and foreign researchers deeply investigate algebra connectivity and Fiedler vectors of a graph, and obtain a lot of significative results.The Laplacian matrix of a mixed graph is a generalization of that of a simple graph. However, when the Laplacian matrix of a mixed graph is nonsingular, the results of latter cannot be simply translated to a mixed graph. Therefore, the singularity is an essential factor which leads the spectral difference between mixed graphs and simple graphs.This thesis mainly deals with the singularity of mixed graphs. Analog to Fiedler using eigenvalues to characterize the structure of a simple graph, we want to characterize the singularity(mainly the least eigenvalue, called singularity of mixed graphs) of Laplacian matrix by the structure property of a mixed graph. We introduce two new parameters of a mixed graph:"the vertex singularity"and"the edge singularity", and use them to describe the singularity of mixed graphs. On the other hand, considering that the Fiedler vectors of a simlpe graph has a beautiful combinatorial property, Fan et.al. extend this property to nonsingular unicyclic mixed graphs and mixed graphs with exactly one nonsingular cycle. In this thesis we extend this property further to mixed graphs with edge singularity being one which contains above two classes of graphs.The thesis is organized as follows: In Chapter 1, we firstly introduce a background of the Lapalacian spectral theory, some concepts and notations used in this thesis, then discuss the research problems together with their developments, and list the main results we obtain in this thesis. In Chapter 2, we introduce the concepts of vertex singularity and edge singularity, and establish the relations of the two parameters and the least eigenvalue of mixed graphs. In Chapter 3, we firstly introduce the structure of eigenvectors of nonsingular unicycle graphs and mixed graphs with exactly one nonsingular cycle, and then we extend this property to mixed graphs with edge singularity being one.