An Algorithm for Determining Isomorphism Using Lexicographic Sorting and the Matrix Inverse

The PageRank algorithm perturbs the adjacency matrix defined by a set of web pages and hyperlinks such that the resulting matrix is positive and row-stochastic. Applying the Perron-Frobenius theorem establishes that the eigenvector associated with the dominant eigenvalue exists and is unique. For some graphs, the PageRank algorithm may yield a canonical isomorph. We propose a ranking method based on the matrix inverse. Since the inverse may not exist, we apply two isomorphismpreserving perturbations, based on the signless Laplacian, to ensure that the resulting matrix is diagonally dominant. By applying the Gershgorin Circle theorem, we know this matrix must have an inverse, namely, a set of vectors unique up to isomorphism. We concatenate sorted rows of the inverse with its unsorted rows, lexicographically sort on the concatenated matrix, and apply the ranking as an induced permutation on the input adjacency matrix. This preliminary report shows IsoRank identifies most random graphs and always terminates in polynomial time, illustrated by the execution run times for a small set of graphs. IsoRank has been applied to dense graphs of as many as 4,000 vertices.