The Laplacian spectrum of a graph

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertexdegrees and by A(G) its adjacency matrix. Then, the Laplacian matrix of G is L(G) = D(G) − A(G). The first and second section of this paper contains introduction and some known results, respectively. The third section is devoted to properties of Laplacian spectrum. The fourth section contains characterization of graphs. The fifth section relates the Laplacian eigenvalues with the graph structure.