  For any $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ the symmetric
decreasing rearrangement $f^*$ satisfies the Polya-Szeg\H{o} inequality $\|
\nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$. The goal of this paper is to
establish analogous results in the discrete setting for graphs satisfying
suitable conditions. We prove that if the edge-isoperimetric problem on a graph
has a sequence of nested minimizers, then this sequence gives rise to a
rearrangement satisfying the Polya-Szeg\H{o} inequality in $L^1$. This shows,
for example, that a specific rearrangement on the grid graph $\mathbb{Z}^2$,
going around the origin in a spiral-like manner, satisfies $\| \nabla
f^*\|_{L^1} \leq \| \nabla f\|_{L^1}$. The $L^{\infty}-$case is implied by an
optimal ordering condition in vertex-isoperimetry. We use these ideas to prove
that the canonical rearrangement on the infinite $d-$regular tree satisfies the
Polya-Szeg\H{o} inequality for all $1 \leq p \leq \infty$.
