Compared with monotherapy, anti-cancer drug combination can provide effective therapy with less toxicity in cancer treatment. Recent studies found that the topological positions of protein modules related to the drugs and the cancer cell lines in the protein-protein interaction (PPI) network may reveal the effects of drugs. However, due to the size of the combinatorial space, identifying synergistic combinations of drugs from PPI network is computationally difficult. To address this challenge, we propose an end-to-end deep learning framework, namely Graph Convolutional Network for Drug Synergy (GraphSynergy), to make synergistic drug combination predictions. GraphSynergy adapts a spatial-based Graph Convolutional Network component to encode the high-order structure information of protein modules targeted by a pair of drugs, as well as the protein modules associated with a specific cancer cell line in the PPI network. The pharmacological effects of drug combinations are explicitly evaluated by their therapy and toxic scores. By introducing an attention component to automatically allocate contribution weights to the proteins, we show the ability of GraphSynergy to capture the pivotal proteins that play a part in both PPI network and biomolecular interactions between drug combinations and cancer cell lines. Experiments on two latest drug combination datasets demonstrate that GraphSynergy outperforms the state-of-the-art in predicting synergistic drug combinations. This study sheds light on using machine learning to discover effective combination therapies for cancer and other complex diseases.