Complex signaling pathways/networks are believed to be responsible for drug resistance in cancer therapy. Drug combinations inhibiting multiple signaling targets within cancer-related signaling networks have the potential to reduce drug resistance. Deep learning models have been reported to predict drug combinations. However, these models are hard to be interpreted in terms of mechanism of synergy (MoS), and thus cannot well support the human-AI based clinical decision making. Herein, we proposed a novel computational model, DeepSignalingFlow, which seeks to address the preceding two challenges. Specifically, a graph convolutional network (GCN) was developed based on a core cancer signaling network consisting of 1584 genes, with gene expression and copy number data derived from 46 core cancer signaling pathways. The novel up-stream signaling-flow (from up-stream signaling to drug targets), and the down-stream signaling-flow (from drug targets to down-stream signaling), were designed using trainable weights of network edges. The numerical features (accumulated information due to the signaling-flows of the signaling network) of drug nodes that link to drug targets were then used to predict the synergy scores of such drug combinations. The model was evaluated using the NCI ALMANAC drug combination screening data. The evaluation results showed that the proposed DeepSignalingFlow model can not only predict drug combination synergy score, but also interpret potentially interpretable MoS of drug combinations.