While there have been many studies on hardware acceleration for deep learning on images, there has been a rather limited focus on accelerating deep learning applications involving graphs. The unique characteristics of graphs, such as the irregular memory access and dynamic parallelism, impose several challenges when the algorithm is mapped to a CPU or GPU. To address these challenges while exploiting all the available sparsity, we propose a flexible architecture called SPA-GCN for accelerating Graph Convolutional Networks (GCN), the core computation unit in deep learning algorithms on graphs. The architecture is specialized for dealing with many small graphs since the graph size has a significant impact on design considerations. In this context, we use SimGNN, a neural-network-based graph matching algorithm, as a case study to demonstrate the effectiveness of our architecture. The experimental results demonstrate that SPA-GCN can deliver a high speedup compared to a multi-core CPU implementation and a GPU implementation, showing the efficiency of our design.