Orthogonal Eigenvector Matrix of the Laplacian

Abstract

The orthogonal eigenvector matrix Z of the Laplacian matrix of a graph with N nodes is studied rather than its companion X of the adjacency matrix, because for the Laplacian matrix, the eigenvector matrix Z corresponds to the adjacency companion X of a regular graph, whose properties are easier. In particular, the column sum vector of Z (which we call the fundamental weight vector w) is, for a connected graph, proportional to the basic vector e_N = (0, 0, . . . , 1), so that more information about the speclics of the graph is contained in the row sum of Z (which we call the dual fundamental weight vector φ). Since little is known about Z (or X), we have tried to understand simple properties of Z such as the number of zeros, the sum of elements, the maximum and minimum element and properties of φ. For the particular class of Erdos-Rényi random graphs, we found that a product of a Gaussian and a super-Gaussian distribution approximates accurately the distribution of Φ_U, a uniformly at random chosen component of the dual fundamental weight vector of Z.