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

Abstract

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.

Some further development on the eigensystem approach for graph isomorphism detection

“…This paper presented such a matrix called adjusted adjacency matrix (AAM). We proved that the AAM can meet that requirement and showed that the eigensystem approach based on the AM [25,29] can be readily migrated to the one based on the AAM. We conducted the computational complexity for the eigensystem approach based on the AAM.…”

“…The new method is called the eigensystem approach. In [29] we further improved this method yet left only one requirement on this method; that is, the AM of a graph must contain at least one distinct eigenvalue.…”

“…Elsewhere, we proposed a method which is based on a combination of eigenvalue and eigenvector, as opposed to relevant methods in the literature which only make use of the eigenvalue [25,29]. This method is called the eigensystem approach.…”

The isomorphism identification of reducible performance for parallel mechanisms

“…Chang et al [ 7 ] used the eigensystem method for mechanism kinematic chain isomorphism identification, and Cubillo et al [ 8 ] commented on mechanism kinematic chain isomorphism identification using adjacency matrices. He et al [ 9 , 10 ] discussed the eigenvalue and eigenvector information of graphs and their validity in detection of graph isomorphism, and then gave some further developments on the eigensystem approach for graph isomorphism detection. The eigensystem approach has been shown to be very effective, but requires that the adjacency matrix must contain at least one distinct eigenvalue.…”

Parallel mechanism character arrays of topology graphs isomorphism identification and creation automatically

“…But when the primary Hamming string fails, the computation of the secondary Hamming string is needed. The method of eigenvectors and eigenvalues of adjacency matrices [16,17] detects isomorphism through computing both the eigenvectors and eigenvalues of the vertex adjacency matrices. Its calculation is complex and it is hard to analyze the topological structure of kinematic chains.…”