Exact polynomial-time graph canonisation and isomorphism testing through comparison of ordered vertex eigenprojections

Motivation Graph canonisation and isomorphism testing representation are fundamental computational problems, whose complexity has remained unsolved to date. This study examines graph eigenprojections, demonstrating that linear-ordering transformations induce canonical properties therein to yield polynomial-time canonisation and isomorphism testing in all undirected graphs. Results This study presents an exact method to identify analogous vertices in isomorphic graphs, through comparison of vertices’ eigenprojection matrices, which are shown to be related by a linear permutation. Systematic perturbation strategies are developed to reduce degeneracy whilst conserving isomorphism, through the addition of characteristically weighted self-loops to analogous vertices. Repeated iterations of analogy testing and perturbation deliver canonical vertex labelling and recovery of isomorphic mappings in time in all graphs. Analytical proofs are provided to support claims and experimental performance is demonstrated in biological and synthetic data, with comparison to a commonly used heuristic algorithm. Availability and Implementation Source code is provided at github.com/robertoshea/graph_isomorphism . Contact [email protected] Supplementary Data. Not applicable.