  We consider a hierarchy of graph invariants that naturally extends the
spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on the
angles formed by the set of standard basis vectors and their projections onto
eigenspaces of the adjacency matrix. We provide a purely combinatorial
characterization of this hierarchy in terms of the walk counts. This allows us
to give a complete answer to F\"urer's question about the strength of his
invariants in distinguishing non-isomorphic graphs in comparison to the
2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan
and Seppelt (SODA 2023). As another application of the characterization, we
prove that almost all graphs are determined up to isomorphism in terms of the
spectrum and the angles, which is of interest in view of the long-standing open
problem whether almost all graphs are determined by their eigenvalues alone.
Finally, we describe the exact relationship between the hierarchy and the
Weisfeiler-Leman algorithms for small dimensions, as also some other important
spectral characteristics of a graph such as the generalized and the main
spectra.
