  Let $p$ be a real number greater number greater than one. Suppose that a
graph $G$ of bounded degree is quasi-isometric with a Riemannian manifold $M$
with certain properties. Under these conditions we will show that the
$p$-harmonic boundary of $G$ is homeomorphic to the $p$-harmonic boundary of
$M$. We will also prove that there is a bijection between the $p$-harmonic
functions on $G$ and the $p$-harmonic functions on $M$.
