Resistance Distance

Abstract

Threshold graphs have an interesting structure and they arise in many areas. We will be particularly interested in the Laplacian eigenvalues of threshold graphs. We first review some basic aspects of the theory of majorization. The majorization between the eigenvalues and the diagonal entries of a symmetric matrix is proved. Majorization for integer vectors is considered and the necessity part of the theorem of Gale and Ryser is proved. Threshold graphs are defined in the next section. It is proved that a graph is threshold if and only if the eigenvalues of its Laplacian are given by the conjugate of its degree sequence. In the final section we first define cographs, which include threshold graphs, and also have the property that the Laplacian eigenvalues are integral. We the obtain some results concerning the effect of a rank one perturbation on the eigenvalues of a symmetric matrix. The results are applied to examine the change in the Laplacian eigenvalues of a graph, when a single edge is added to the graph.