A Spectral Assignment Approach for the Graph Isomorphism Problem

In this paper, we propose a heuristic for the graph isomorphism problem that is based on the eigendecomposition of the adjacency matrices. It is well known, that the eigenvalues of the adjacency matrices of isomorphic graphs need to be identical. However, two graphs GA and GB can be isospectral but non-isomorphic. If the graphs possess repeated eigenvalues, which typically correspond to graph symmetries, finding isomorphisms is much harder. By repeatedly perturbing the adjacency matrices, it is possible to break symmetries of the graphs without changing the isomorphism and to assign vertices of GA to vertices of GB, provided that an admissible assignment exists. This heuristic approach can be used to construct a permutation which transforms GA into GB if the graphs are isomorphic, or to show that no isomorphism exists.