Wave equations for graphs and the edge-based Laplacian
The authors study what appears to be a new type of wave equation on graphs. This wave equation (1) involves a reasonable analogue of \(u_{tt}=\Delta u\) in analysis, (2) has ``finite speed of 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 This wave equation requires some of the notions of ``calculus of graphs'' from an unpublished preprint of the authors. As one of the examples, the authors give a simple proof of a relation between distances of sets, their sizes, and the first nonzero edge-based eigenvalue -- which can be better (or worse) than that of [\textit{F. R. K. Chung, V. Faber, T. A. Manteuffel}, SIAM J. Discrete Math. 7 (3), 443--457 (1994; Zbl 0808.05072)].
scientific article; zbMATH DE number 2138444