  For Paley-Wiener functions on weighted combinatorial finite or infinite
graphs we develop a weighted sampling theory in which samples are defined as
inner products with weight functions (measuring devices). Three reconstruction
methods are suggested. The first two of them are using language of dual Hilbert
frames and the so-called frame algorithm respectively. The third one is using
the so-called weighted variational interpolating splines which are constructed
in the setting of combinatorial graphs. This development requires a new set of
Poincar\'e-type inequalities which we prove for functions on combinatorial
graphs.
