Sobolev and Hardy-Sobolev spaces on graphs

Let $\mathrm{\Gamma }$ be a graph. Under suitable geometric assumptions on $\mathrm{\Gamma }$, we give several equivalent characterizations of Sobolev and Hardy-Sobolev spaces on $\mathrm{\Gamma }$, in terms of maximal functionals, Haj{\l} asz type functionals or atomic decompositions. As an application, we study the boundedness of Riesz transforms on Hardy spaces on $\mathrm{\Gamma }$. This gives the discrete counterpart of the corresponding results on Riemannian manifolds.