This paper proposes a family of permutation-invariant graph embeddings, generalizing the Skew Spectrum of graphs of Kondor & Borgwardt (2008). Grounded in group theory and harmonic analysis, our method introduces a new class of graph invariants that are isomorphism-invariant and capable of embedding richer graph structures - including attributed graphs, multilayer graphs, and hypergraphs - which the Skew Spectrum could not handle. Our generalization further defines a family of functions that enables a trade-off between computational complexity and expressivity. By applying generalization-preserving heuristics to this family, we improve the Skew Spectrum's expressivity at the same computational cost. We formally prove the invariance of our generalization, demonstrate its improved expressiveness through experiments, and discuss its efficient computation.

Graphs are a powerful way to represent complex systems, from molecules and social networks to financial connections. But teaching computers to compare graphs is surprisingly tricky. Imagine two subway maps for the same city: the stations and routes are identical, but laid out differently. To a person, it's clearly the same network. To a computer, they can look completely different. The challenge is finding a machine representation of a graph that expresses the connections unambiguously, regardless of how the routes are laid out.We address this problem using abstract mathematical tools for symmetry and signal processing - specifically group theory, representation theory, and Fourier analysis. Building on a classical method called the Skew Spectrum, we create a new family of graph "fingerprints" that are invariant to graph reordering. Unlike earlier techniques, ours can also handle more complex systems - like graphs with attributes, multiple layers, or richer types of connections. Our approach is flexible: it lets users trade off between computational cost and representational power.This result makes it easier for machines to compare complex networks in the real world, potentially enhancing graph neural networks and applications in drug discovery, fraud detection, and social science.