  In this paper, we study the wave equation on infinite graphs. On one hand, in
contrast to the wave equation on manifolds, we construct an example for the
non-uniqueness for the Cauchy problem of the wave equation on graphs. On the
other hand, we obtain a sharp uniqueness class for the solutions of the wave
equation. The result follows from the time analyticity of the solutions to the
wave equation in the uniqueness class. In the last part, we extend the result
to a wide class of linear evolution equations.
