Graph characteristics from the Schrödinger operator

Abstract

In this paper, we show how the Schrödinger operator may be applied to the problem of graph characterization. The motivation is the similarity of the Schrödinger equation to the heat difussion equation, and the fact that the heat kernel has been used in the past for graph characterization. Our hypothesis is that due to the quantum nature of the Schrödinger operator, it may be capable of providing richer sources of information than the heat kernel. Specifically the possibility of complex amplitudes with both negative and positive components, allows quantum interferences which strongly reflect symmetry patterns in graph structure. We propose a graph characterization based on the Fourier analysis of the quantum equivalent of the heat flow trace. Our experiments demonstrate that this new method can be succesfully applied to characterize different types of graph structures. © 2013 Springer-Verlag.