Calculus on Graphs

TL;DR

This paper introduces a calculus framework on graphs that connects graph theory with analysis, enabling new PDEs, improved bounds, and transfer of nonlinear techniques to graph structures.

Contribution

It develops a novel calculus on graphs that facilitates analysis-based methods and improves bounds in graph theory and analysis.

Findings

New wave equation on graphs derived from Laplacian

Improved diameter/eigenvalue bounds in graph theory

Transfer of nonlinear p-Laplacian techniques to graphs

Abstract

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; this wave equation gives rise to a partial improvement on the Chung-Faber-Manteuffel diameter/eigenvalue bound in graph theory, and the Chung-Grigoryan-Yau and (in a certain case) Bobkov-Ledoux distance/eigenvalue bounds in analysis. Our framework also allows most techniques for the non-linear p-Laplacian in analysis to be easily carried over to graph theory.