Poincar\'e and plancherel-polya inequalities in harmonic analysis on weighted combinatorial graphs

TL;DR

This paper establishes Poincaré and Plancherel-Polya inequalities for weighted graphs, linking geometric and spectral properties, and demonstrates their use in sampling and data analysis on graphs.

Contribution

It introduces explicit inequalities with geometric constants for weighted graphs, connecting spectral and geometric graph properties, and applies these to sampling and data analysis.

Findings

Derived explicit Poincaré inequalities with geometric constants.

Established relations between geometric and spectral properties of Laplacian.

Demonstrated Shannon-type sampling on line graphs.

Abstract

We prove Poincar\'e and Plancherel-Polya inequalities for weighted {\ell}p -spaces on weighted graphs in which the constants are explicitly expressed in terms of some geometric characteristics of a graph. We use Poincar\'e type inequality to obtain some new relations between geometric and spectral properties of the combinatorial Laplace operator. Several well known graphs are considered to demonstrate that our results are reasonably sharp. The Plancherel-Polya inequalities allow for application of the frame algo- rithm as a method for reconstruction of Paley-Wiener functions on weighted graphs from a set of samples. The results are illustrated by developing Shannon- type sampling in the case of a line graph. Our work has potential applications to data mining and learning theory on graphs.