Poincaré and Plancherel-Polya Inequalities in Harmonic Analysis on Weighted Combinatorial Graphs

Abstract. We prove Poincaré and Plancherel–Polya inequalities for weighted p-spaces on weighted graphs in which the constants are explicitly expressed in terms of some geometric characteristics of a graph. We use a Poincaré-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 algorithm 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.