Let $\Gamma$ be a graph with a weight $\sigma$. Let $d$ and $\mu$ be the distance and the measure associated with $\sigma$ such that $(\Gamma, d, \mu)$ is a doubling space. Let $p$ be the natural reversible Markov kernel associated with $\sigma$ and $\mu$  and $P$ the associated
operator defined by $Pf(x) = \sum_{y} p(x, y)f(y)$. Denote by $L=I-P$ the discrete Laplacian on $\Gamma$. 
In this talk we develop the theory of Hardy spaces associated to the discrete Laplacian $H^p_L$ for $0<p\leq 1$. We then obtain boundedness of certain singular integrals on $\Gamma$ such as square functions, spectral multipliers and Riesz transforms on the Hardy spaces $H^p_L$. This is joint work with The Anh Bui.