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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 conditionson $\Gamma$ 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 conditionsare different for $p<2$ and $p>2$. The proofs rely on recent techniquesdeveloped to handle operators beyond the class of Calder\'on-Zygmund operators.For our purpose, we also prove Littlewood-Paley inequalities and interpolationresults for Sobolev spaces in this context, which are of independent interest.

Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs