Graphs Laplace Eigenvalue

The Laplacian eigenvalue is an active and important subject in Algebra Graph Theory. This thesis presents a systematic research on two types of graphs, that is undirected graph and mixed graph. Using the matrix knowledge, we give the following conclutions.Firstly, we give a new upper bound of a mixed graph′s largest Laplacian eigenvalue, which is more precise. And also we give a lower bound of a mixed connected graph′s largest Laplacian eigenvalue.Secondly, we contribute to some principles about Laplacian eigenvalue and eigenvector of mixed graph. These principles illustrate the effects on the Laplacian eigenvalues and eigenvectors of mixed graph by adding edges, coalescing vertexes, etc. In addition, we give the Laplacian eigenvalues of a graph obtained by adding a perfect matching between two similar graphs, and also give the Laplacian eigenvalues of a graph obtained by adding a perfect matching among one graph.In the last, we give a new upper bound of the algebraic connectivity about the dominate number.