We study the boundedness on the Hardy spaces of spectral multiplier operators associated with the discrete Laplacian on a weighted graph. We assume that the graph satisfies the doubling volume property and a Poincaré inequality. We prove that there is , depending on the geometry of the graph, such that if the multiplier satisfies a condition similar to the one we have in the classical Hörmander multiplier theorem, then the corresponding operator is bounded on , .