On graph isomorphism and graph automorphism

Abstract

The problem of graph isomorphism, graph automorphism and a unique graph ID is considered. A new approach to the solution of these problems is suggested. The method is based on the spectral decompositionA =ΣiλiKiof the adjacency matrixA. This decomposition is independent of the particular labeling of graph vertices, and using this decomposition one can formulate an algorithm to derive a canonical labeling of the corresponding graphG. Since the spectral decomposition uniquely determines the adjacency matrixA and hence graphG, the obtained canonical labeling can be used in order to derive a unique graph ID. In addition, if the algorithm produces several canonical labelings, all these labelings and only these labelings are connected by the elements of the graph automorphism groupG. In this way, one obtains all elements of this group. Concerning graph isomorphism, one can use a unique graph ID obtained in the above way. However, the algorithm to decide whether graphG andG′ are isomorphic can be substantially improved if this algorithm is based on the direct comparison between spectral decompositions of the corresponding adjacency matricesA andA′.