Eigenvalues of the normalized Laplacian

A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices have been well studied for graphs. A newer matrix is the normalized Laplacian L = D−1/2LD−1/2 which can be thought of as a cross in between these two other matrices and is related to the transition matrix for random walks. If we have the matrix then we completely understand the graph since it represents the graph. But sometimes it might be too much to hope to have the matrix as it requires a large amount of memory to store the matrix (i.e., a graph on n vertices can require something on the order of n to store the matrix, for a graph with billions of nodes, such as the internet this becomes prohibitive). As an alternative we might study the eigenvalues of the matrix. These eigenvalues give us some useful information about the matrix which can be translated into useful information about the graph. The area of spectral graph theory ties together the eigenvalues of the matrix with the structure of a graph. The eigenvalues of the adjacency and combinatorial Laplacian have been well studied. On the other hand the eigenvalues of the normalized Laplacian have only recently been studied, most importantly in the work of Fan Chung [23].