A Spectral Assignment Approach for the Graph Isomorphism Problem

TL;DR

This paper introduces a spectral method for the graph isomorphism problem that uses eigenvalues and eigenvectors of adjacency matrices, combined with perturbations and linear assignment techniques, to detect isomorphisms.

Contribution

It develops a novel spectral assignment algorithm that extends to unambiguous graphs and handles symmetries through matrix perturbations and eigenpolytope properties.

Findings

Effective in distinguishing isomorphic graphs with unique spectra

Handles symmetries by perturbing adjacency matrices

Provides a heuristic for constructing graph isomorphisms

Abstract

In this paper, we propose algorithms for the graph isomorphism (GI) problem that are based on the eigendecompositions of the adjacency matrices. The eigenvalues of isomorphic graphs are identical. However, two graphs $ G_A $ and $ G_B $ can be isospectral but non-isomorphic. We first construct a graph isomorphism testing algorithm for friendly graphs and then extend it to unambiguous graphs. We show that isomorphisms can be detected by solving a linear assignment problem. If the graphs possess repeated eigenvalues, which typically correspond to graph symmetries, finding isomorphisms is much harder. By repeatedly perturbing the adjacency matrices and by using properties of eigenpolytopes, it is possible to break symmetries of the graphs and iteratively assign vertices of $ G_A $ to vertices of $ G_B $, provided that an admissible assignment exists. This heuristic approach can be used to construct a permutation which transforms $ G_A $ into $ G_B $ if the graphs are isomorphic. The methods will be illustrated with several guiding examples.