Calculus on Graphs

The purpose of this paper is to develop a “calculus” on graphs that allows graph theory to have new connections to analysis. For example, our framework gives rise to many new partial differential equations on graphs, most notably a new (Laplacian based) wave equation (see [FTa, FTc]); this wave equation gives rise to a partial improvement on the Chung-Faber-Manteuffel diameter/eigenvalue bound in graph theory (see [CFM94]), and the Chung-Grigoryan-Yau and (in a certain case) Bobkov-Ledoux distance/eigenvalue bounds in analysis (see [CGY96, CGY97, BL97]). Our framework also allows most techniques for the non-linear p-Laplacian in analysis to be easily carried over to graph theory (see [FTb]).