The Laplacian Spectra Of Some Graphs

The spectral theory in graphs is an important and active field, we can study the eigenvalues of some matrixes to characterize the property of graphs, which is an important issue in the spectral theory, especially about the adjacency matrix and Laplacian matrix. In this paper, we determine the extremal graphs corresponding to the Laplacian spectral radius in the class of bipartite graphs, and we discuss the Laplacian spectra of transformation graphs ofγ-regular graph.Let Bnκbe the class of bipartite graphs with n vertices andκcut edges. This paper presents the extremal graphs with the first and the second largest Laplacian spectral radius among all graphs in Bnκ. The bounds of the Laplacian spectral radius of these extremal graphs are also obtained.The Laplacian spectrum of transformation graph is discussed in the paper. The Laplacian characteristic polynomials of eight transformation graphs of a regular graph G is presented in terms of the Laplacian spectrum of G.