The analysis of 3D point clouds has diverse applications in robotics, vision and graphics. Processing them presents specific challenges since they are naturally sparse, can vary in spatial resolution and are typically unordered. Graph-based networks to abstract features have emerged as a promising alternative to convolutional neural networks for their analysis, but these can be computationally heavy as well as memory inefficient. To address these limitations we introduce a novel Multi-level Graph Convolution Neural (MLGCN) model, which uses Graph Neural Networks (GNN) blocks to extract features from 3D point clouds at specific locality levels. Our approach employs precomputed graph KNNs, where each KNN graph is shared between GCN blocks inside a GNN block, making it both efficient and effective compared to present models. We demonstrate the efficacy of our approach on point cloud based object classification and part segmentation tasks on benchmark datasets, showing that it produces comparable results to those of state-of-the-art models while requiring up to a thousand times fewer floating-point operations (FLOPs) and having significantly reduced storage requirements. Thus, our MLGCN model could be particular relevant to point cloud based 3D shape analysis in industrial applications when computing resources are scarce.