Gaussian Wave Packet on a Graph

Abstract

The wave kernel provides a richer and potentially more expressive means of characterising graphs than the more widely studied wave equation. Unfortunately the wave equation whose solution gives the kernel is less easily solved than the corresponding heat equation. There are two reasons for this. First, the wave equation can not be expressed in terms of the familiar node-based Laplacian, and must instead be expressed in terms of the edge-based Laplacian. Second, the eigenfunctions of the edge-based Laplacian are more complex than those of the node-based Laplacian. This paper presents the solution of a wave equation on a graph. Wave equation provides an interesting alternative to the heat equation defined using the Edge-based Laplacian. This provides the prerequisites for deeper analysis of graphs and their characterisation. For instance it potentially allows the study of non-dispersive solutions or solitons. In this paper we give a complete solution of the wave equation for a Gaussian wave packet. To simulate the equation on a graph, we assume the initial distribution be a Gaussian wave packet on a single edge of the graph. We show the evolution of this Gaussian wave packet with time on some synthetic graphs.