To explore and enrich the potential of graphene-based neuromorphic computing, we propose a reconfigurable graphene-based Spiking Neural Network (SNN) architecture and a training methodology for initial synaptic weight values determination. The proposed graphene-based platform is flexible, comprising a programmable synaptic array which can be configured for different initial synaptic weights and plasticity functionalities and a spiking neuronal array, onto which various neural network structures can be mapped according to the application requirements and constraints. To reconfigure the proposed graphene-based platform for a practical application, an SNN topology tailored for the application and an initial SNN state (initial synaptic weights, plasticity type), which can be determined by proposed training methodology are required. To demonstrate the validity of the synaptic weights training methodology and the suitability of the proposed SNN architecture for practical utilization, we consider applications, i.e., character recognition and edge detection. In each case, the graphene-based platform is configured as per the application tailored SNN topology and initial state and SPICE simulated to evaluate its reaction to the applied input stimuli. For the first application, a 2-layer SNN with 30 neurons is used to reconfigure the proposed graphene-based architecture and perform character recognition for 5 vowels, i.e., "A", "E", "I", "O", and "U" variations. Our simulation indicates that the graphene-based SNN can achieve up to 94.5 % recognition accuracy for the considered test datasets, which is comparable with the one delivered by a functionally equivalent Artificial Neural Network (ANN). Further, we reconfigure the architecture for a 3-layer 13 neurons SNN to perform edge detection on 2 grayscale images, Lena and Cameraman. SPICE simulation results indicate that the edge extraction results are close agreement with the one produced by classical edge detection operators, i.e., Canny, Roberts, Sobel, and Prewitt, in terms of visual perception, Peak Signal-to-Noise Ratio (PSNR), and Mean Squared Error (MSE). Our results demonstrate that the graphene SNN platform is able to properly perform character recognition and edge detection tasks, which suggests the feasibility and flexibility of the proposed approach for various application purposes. Moreover, the utilized graphene-based synapses and neurons operate at low supply voltage (200 mV), consume low energy per spike for both neuron (43 pJ and 5.2 × 10 −7 pJ at 200 Hz and 20 GHz spike frequency scale, respectively) and synapse (5.1 pJ and 6.0 × 10 −8 pJ at 200 Hz and 20 GHz spike frequency scale, respectively), and a graphene-based synapse occupies an active area of ≈45 nm 2 (2 GNR devices) and a neuron an active area of ≈176 nm 2 (6 GNR devices), which are desired properties for large-scale energy-efficient implementations.