On The Eigenvalues Distribution Of Mixed Graphs

The spectral theory of mixed graphs is a new topic of spectral graph theory. Since it is more difficult to study the Laplacian matrix of a mixed graphs than that of simple graphs, there will be more new problems and new challenges in this field. The thesis mainly deal with the eigenvalues distribution of nonsingular mixed graphs.In Chapter 1, besides introducing a background of the spectral theory of graphs, some concepts and notations, we mainly discuss the research problems, their developments, and some results relative to this tlaesis. In addition, we list the main results obtained in the following chapters.In Chapter 2. we first estabish some reIations between the eigenvalues and diameter, matching number of mixed graphs, which extends some known results of simple graphs to mixed graphs. In Section 2.2, we completely characterize the nonsingular mixed graphs with exactly one and with exactly two eigenvalues greater than 2, respectively.In Chapter 3, we continue the work in Chapter 2, and characterize two classes of nonsingular mixed graphs with at most three eigenvalues greater than 2. Firstly we estabish the relation between the eigenvalues and the number of the quasi-pendent vertices of a mixed graphs, which extends another known result of simple graphs to mixed graphs. In the remaining two sections of this chapter, we characterize the nonsingular unieyelic mixed graphs and the nonsingular bipartite mixed graphs containing cycle C₆ with at most three eigenvalues greater than 2, respectively.