Wave Equations for Graphs and the Edge-based Laplacian

The main goal of this paper is to develop a “wave equation” for graphs that is very similar to the wave equation utt = ∆u in analysis. Whenever this type of wave equation is involved in a result in analysis, our graph theoretic wave equation seems likely to provide the tool to link the result in analysis to an analogous result in graph theory. Traditional graph theory defines a Laplacian, ∆, as an operator on functions on the vertices. This gives rise to a wave equation utt = −∆u (since graph theory Laplacians are positive semidefinite). However, this wave equation fails to have a “finite speed of wave propagation”. In other words, if u = u(x, t) is a solution, we may have u(x, 0) = 0 for all vertices x within a distance d > 0 to a fixed vertex, x0, without having u(x0, ) vanishing for any > 0. As such, this graph theoretic wave equation cannot link most results in analysis involving the wave equation to a graph theoretic analogue. In this paper we study what appears to be a new type of wave equation on graphs. This wave equation (1) involves a reasonable analogue of utt = ∆u in analysis, (2) has “finite speed of wave propagation” and many other basic properties shared by its analysis counterpart, and (3) seems to be a good vehicle for translating results in analysis to those in graph theory, and vice versa. This wave equation cannot be expressed in the language of traditional graph theory; it requires some of the notions of “calculus on graphs” in [FT99]. It does, however, have a simple physical interpretation—namely, the edges are taut strings, fused together at the vertices. And in fact, the type of Laplacian we use has appeared in the physics literature as the “limiting case” of a “quantum wire” (see [Hur00, RS01, KZ01] for example); but our type of development of the wave equation and its applications to graph theory seem to have escaped the interest of physicists.