Let p be a real number greater than one and let G be a connected graph of bounded degree. In this paper we introduce the p-harmonic boundary of G. We use this boundary to characterize the graphs G for which the constant functions are the only p
Nearby in the stack
-harmonic functions on
G
. It is shown that any continuous function on the
p
-harmonic boundary of
G
can be extended to a function that is
p
-harmonic on
G
. Some properties of this boundary that are preserved under rough-isometries are also given. Now let
Γ
be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced
ℓp
-cohomology of
Γ
in terms of the cardinality of its
p
-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on