  Let $\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$
the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by
$\nabla$ the discrete gradient. We give necessary and/or sufficient conditions
on $\Gamma$ in order to compare $\Vert \nabla f \Vert_{p}$ and $\Vert
(I-P)^{1/2}f \Vert_{p}$ uniformly in $f$ for $1<p<+\infty$. These conditions
are different for $p<2$ and $p>2$. The proofs rely on recent techniques
developed to handle operators beyond the class of Calder\'on-Zygmund operators.
For our purpose, we also prove Littlewood-Paley inequalities and interpolation
results for Sobolev spaces in this context, which are of independent interest.
