On The Third Largest Laplacian Eigenvalue Of A Graph

In graph theory,there are all kinds of matrices introduced to study the properties of the graph.It is our main problems that how to use the algebraic properties of matrices(especially eigenvalues)to reflect the properties of graphs.In the study we found that the eigenvalues of the Laplacian matrix have a close relation to numerous graph invariants.Thus,they are more natural and more important than the eigen-values of the adjacency matrix.Currently the eigenvalues of the Laplacian matrix becomes increasingly aroused more and more people's concern and is a hot issue in the study of graph theory.In this paper,the Laplacian eigenvalue is studied and the main contents are as follows:1.In this paper,it introduces the background and significance of Laplacian eigenvalue and studys the Laplacian characteristic polynomial.2.In this paper,it summarizes the Laplacian eigenvalue of graph.For example,the Laplacian spectral radius,the second largest Laplacian eigenvalue,the algebraic connectivity and so on.3.In this paper,it respectively studied the graph that their third largest Laplacian eigenvalue no more than 2.Finally,it gave all the connected graph of the third largest Laplacian eigenvalue no more than 2.