Stability of the eigenvalues of graphs

Abstract

The spectra of graphs has been widely used to characterise and extract information from their structures. Applications include matching, segmentation and indexing. One of the key questions about this approach is the stability and representational power of the spectrum under changes in the graphs. There is also a wide variety of graph matrix representations from which the spectrum can be extracted. In this paper we discuss the issue of stability of various graph representation methods and compare five main graph representations; the adjacency matrix, combinatorial Laplacian, normalized Laplacian matrix, heat kernel and path length distribution matrix. We show that the Euclidean distance between spectra tracks the edit distance over a wide range of edit costs, and we analyse the stability of this relationship. We then use the spectra to match and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates. © Springer-Verlag Berlin Heidelberg 2005.