Some further development on the eigensystem approach for graph isomorphism detection

Many science and engineering problems can be represented by a network, a generalization of which is a graph. Examples of the problems that can be represented by a graph include: cyclic sequential circuit, organic molecule structures, mechanical structures, etc. The most fundamental issue with these problems (e.g., designing a molecule structure) is the identification of structure, which further reduces to be the identification of graph. The problem of the identification of graph is called graph isomorphism. The graph isomorphism problem is an NP problem according to the computational complexity theory. Numerous methods and algorithms have been proposed to solve this problem. Elsewhere we presented an approach called the eigensystem approach. This approach is based on a combination of eigenvalue and eigenvector which are further associated with the adjacency matrix. The eigensystem approach has been shown to be very effective but requires that a graph must contain at least one distinct eigenvalue. The adjacency matrix is not shown sufficiently to meet this requirement. In this paper, we propose a new matrix called adjusted adjacency matrix that meets this requirement. We show that the eigensystem approach based on the adjusted adjacency matrix is not only effective but also more efficient than that based on the adjacency matrix.