  We study the spectral aspects of the graph limit theory. We give a
description of graphon convergence in terms of converegnce of eigenvalues and
eigenspaces. Along these lines we prove a spectral version of the strong
regularity lemma. Using spectral methods we investigate group actions on
graphons. As an application we show that the set of isometry invariant graphons
on the sphere is closed in terms of graph convergence however the analogous
statement does not hold for the circle. This fact is rooted in the
representation theory of the orthogonal group.
