Eigenvalue and Eigenvector Information of Graphs and Their Validity in Detection of Graph Isomorphism

Detection of graph isomorphism (GI) has been widely used in many fields in science and engineering. Currently, a potential application of GI detection could be in molecular structure design for microelectromechanical systems and nano-systems. In this paper, we discuss the relationship between graphs and their eigenvalues as well as unique eigenvectors. We prove that the graphs having all distinct eigenvalues are isomorphic if and only if they have the same graph spectrum and the equivalent eigenvectors. The graphs having coincident eigenvalues might be isomorphic if they have the same graph spectrum and the equivalent unique eigenvectors. Further, a convergent recursive procedure is given to subdivide a group-to-group mapping once appeared in the graphs having coincident eigenvalues to seek potential one-to-one mappings so as to determining if the graphs are isomorphic.