In this paper, we consider the problem of representing and recovering graph signals with a nonlinear measurement model. We propose a two-stage graph signal processing (GSP) framework. First, a GSP representation is obtained by finding the graph filter that best approximates the known measurement function. The new GSP representation enables performing tractable operations over graphs, as well as gaining insights into the signal graph-frequency contents. Then, we formulate the signal recovery problem under the smoothness constraint and derive a regularized least-squares (LS) estimator, which is obtained by applying the inverse of the approximated graph filter on the nonlinear measurements. In the second part of this paper, we investigate the proposed recovery and representation approach for the special case of graph signals that are influenced by the differences between vertex values only. Difference-based graph signals arise, for example, when modeling power signals as a function of the voltages in electrical networks. We show that any difference-based graph signal corresponds to a filter that lacks the zero-order filter coefficient, and thus, these signals can be recovered up to a constant by the regularized LS estimator. In our simulations, we show that for the special case of state estimation in power systems the proposed GSP approach outperforms the state-of-the-art estimator in terms of total variation.