  Graph spectra have been successfully used to classify network types, compute
the similarity between graphs, and determine the number of communities in a
network. For large graphs, where an eigen-decomposition is infeasible,
iterative moment matched approximations to the spectra and kernel smoothing are
typically used. We show that the underlying moment information is lost when
using kernel smoothing. We further propose a spectral density approximation
based on the method of Maximum Entropy, for which we develop a new algorithm.
This method matches moments exactly and is everywhere positive. We demonstrate
its effectiveness and superiority over existing approaches in learning graph
spectra, via experiments on both synthetic networks, such as the
Erd\H{o}s-R\'{e}nyi and Barab\'{a}si-Albert random graphs, and real-world
networks, such as the social networks for Orkut, YouTube, and Amazon from the
SNAP dataset.
