In this paper we propose a generic graphene-based Spiking Neural Network (SNN) architecture for pattern recognition and the associated weight values initialization methodology. The SNN has a Winner-Takes-All 3-layer structure and exhibits tuneable recognition accuracy by exploiting interpatterns similarity/dissimilarity. To demonstrate the capabilities of our proposal we present an SNN instance tailored for low resolution MNIST handwritten digits recognition and evaluate its recognition accuracy by means of SPICE simulations. 2 voltage levels are initially utilized for synaptic weight values representation and the recognition accuracy varies from 75.8% to 99.2%, which, together with its compactness and energy efficient (pJ range/spike), suggests that our approach has great potential for edge device implementations.