We consider functions on a weighted combinatorial graph G (finite or countable) whose evolution in time −∞ < t < ∞ is governed by the Schro ̈dinger type equation ∂g(t, v)/∂t = i∆g(t, v), v ∈ V (G), with the combinatorial Laplace operator on the right side. Two Shannon-type sampling theorems are proved which imply that if the initial data g(0, v) is a Paley-Wiener function of bandwidth ≤ ω then the solution g(t, v) for all t ∈ (−∞,∞), v ∈ V(G) can be perfectly reconstructed from the values of g on a set K ×S where K is a sufficiently dense set of equally spaced real numbers and S is a sampling set for the space of Paley-Wiener function of bandwidth ≤ ω . We also consider an inverse problem of reconstructing the initial function g(0,·) from a single sample g(T, ·).