Spectral classes of regular, random, and empirical graphs

Abstract: We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families that share the same asymptotic behavior of the spectrum so that the distance of two graphs from the same family is bounded by $\mathcal{O}(1/n)$ in terms of size $n$ of their vertex sets. Numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity.