Representing and exploiting multivariate signals requires capturing relations between variables, which we can represent by graphs. Graph dictionaries allow to describe complex relational information as a sparse sum of simpler structures, but no prior model exists to infer such underlying structure elements from data. We define a novel Graph-Dictionary signal model, where a finite set of graphs characterizes relationships in data distribution as filters on the weighted sum of their Laplacians. We propose a framework to infer the graph dictionary representation from observed node signals, which allows to include a priori knowledge about signal properties, and about underlying graphs and their coefficients. We introduce a bilinear generalization of the primal-dual splitting algorithm to solve the learning problem. We show the capability of our method to reconstruct graphs from signals in multiple synthetic settings, where our model outperforms popular baselines. Then, we exploit graph-dictionary representations in an illustrative motor imagery decoding task on brain activity data, where we classify imagined motion better than standard methods relying on many more features. Our graph-dictionary model bridges a gap between sparse representations of multivariate data and a structured decomposition of sample-varying relationships into a sparse combination of elementary graph atoms.