An Efficient and Accurate Graph Isomorphism Test using Graph Spectra

Abstract

Graph isomorphism test is an important and essential task for all methodologies dealing with graphs such as graph mining, graph query, and etc. However, it is a computationally hard problem, and has been a bottleneck of various applications using graphs. In most cases, it needs the cost of O(n!). In this paper, we propose a method to check graph isomorphism using graph spectra and the other graph invariants. A graph spectrum of a graph is ordered eigenvalues of a symmetric matrix representing the graph, and is known to be invariant over any representations of an identical graph. Our method approximately checks graph isomorphism of two graphs by comparing their graph spectra as well as the other graph invariants. This approach could be more efficient compared with existing test because the computational cost of graph spectrum for an n×n symmetric matrix is O(n2). Through experimental evaluations, we will show the graph spectra of the signless Laplacian matrix of a given graph and that of the line graph of the original graph are mutually complementary, and the use of both the spectra could achieve an approximate but highly accurate and highly efficient graph isomorphism test.