Harmonic Labeling of Graphs

Which graphs admit an integer value harmonic function which is injective and surjective onto $\mathbb{Z}$? Such a function, which we call harmonic labeling, is constructed when the graph is the ${\mathbb{Z}}^2$ square grid. It is shown that for any finite graph $G$ containing at least one edge, there is no harmonic labeling of $G\times \mathbb{Z}$.