  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.
