Relation between the Eigenvalues of a Graph and Its Line Graph or Laplacian Eigenvalues

Graph spectral theory is not only an important field in graph theory,but also special active direction.It has been proved that computing eigenvalues are quite complicated,but can study between the relationships about the spectral of the different definitions to determine the bounds of the eigenvalues.By using some characters of the real symmetrical matrix and semi-positive definite matrix,some relations on the adjacent spectrum of the simple undirected graph G and its line graph Gl are studied in this paper,the past results are generalized.At the same time,two relations between the adjacent spectrum of a graph and its Laplacian spectrum are also obtained in the paper.This is valuable to guiding the bounds of the eigenvalues estimation.