  The Agmon estimate shows that eigenfunctions of Schr\"odinger operators, $
-\Delta \phi + V \phi = E \phi$, decay exponentially in the `classically
forbidden' region where the potential exceeds the energy level $\left\{x: V(x)
> E \right\}$. Moreover, the size of $|\phi(x)|$ is bounded in terms of a
weighted (Agmon) distance between $x$ and the allowed region. We derive such a
statement on graphs when $-\Delta$ is replaced by the Graph Laplacian $L =
D-A$: we identify an explicit Agmon metric and prove a pointwise decay estimate
in terms of the Agmon distance.
