Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms

TL;DR

This paper introduces graph equivalence classes, isomorphic and Jordan, to analyze and optimize spectral projector-based graph Fourier transforms, enabling computational efficiency and basis-invariant spectral analysis.

Contribution

It defines and explores isomorphic and Jordan equivalence classes, providing new insights into transform invariance and computational reduction in graph Fourier analysis.

Findings

Isomorphic classes relate transforms to node label permutations.

Jordan classes allow basis-invariant spectral component ordering.

Methods to exploit classes reduce computation time.

Abstract

We define and discuss the utility of two equivalence graph classes over which a spectral projector-based graph Fourier transform is equivalent: isomorphic equivalence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent up to a permutation on the node labels. Jordan equivalence classes permit identical transforms over graphs of nonidentical topologies and allow a basis-invariant characterization of total variation orderings of the spectral components. Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed.