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Calculus on Graphs

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arxiv cs/0408028 v1 pith:QTMEUB3P submitted 2004-08-12 cs.DM math.CO

Joel Friedman , Jean-Pierre Tillich This is my paper · ORCID
classification cs.DMmath.CO
keywords analysisgraphgraphstheoryallowscalculuseigenvalueequation
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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.

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    Nonexistence of all very weak solutions to semilinear parabolic and hyperbolic inequalities on metric graphs under weighted space-time volume growth conditions on the potential, via a new pseudo-metric and coupled/sep...

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