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 StreamGCN for accelerating Graph Convolutional Networks (GCN), the core computation unit in deep learning algorithms on graphs. The architecture is specialized for streaming processing of many small graphs for graph search and similarity computation. The experimental results demonstrate that StreamGCN can deliver a high speedup compared to a multi-core CPU and a GPU implementation, showing the efficiency of our design.