THE LAPLACIAN SPECTRUM OF A GRAPH .2.

Let G be a graph. Denote by D(G) the diagonal matrix of its vertex deg rees and by A(G) its adjacency matrix. Then L(G) = D(G) - A(G) is the Laplacian matrix of G. The first section of this paper is devoted to p roperties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers. The second section relates the degree sequence and the Laplacian spectrum through majorization. The third section introduces the notion of a d-cluster, using it to bound the multiplicity of d in the spectrum Of L(G).