Graph Convolutional Networks (GCNs) have shown to be a powerful tool for analyzing graph-structured data. Most of previous GCN methods focus on learning a good node representation by aggregating the representations of neighboring nodes, whereas largely ignoring the edge information. Although few recent methods have been proposed to integrate edge attributes into GCNs to initialize edge embeddings, these methods do not work when edge attributes are (partially) unavailable. Can we develop a generic edge-empowered framework to exploit node-edge enhancement, regardless of the availability of edge attributes? In this paper, we propose a novel framework EE-GCN that achieves node-edge enhancement. In particular, the framework EE-GCN includes three key components: (i) Initialization: this step is to initialize the embeddings of both nodes and edges. Unlike node embedding initialization, we propose a line graph-based method to initialize the embedding of edges regardless of edge attributes. (ii) Feature space alignment: we propose a translation-based mapping method to align edge embedding with node embedding space, and the objective function is penalized by a translation loss when both spaces are not aligned. (iii) Node-edge mutually enhanced updating: node embedding is updated by aggregating embedding of neighboring nodes and associated edges, while edge embedding is updated by the embedding of associated nodes and itself. Through the above improvements, our framework provides a generic strategy for all of the spatial-based GCNs to allow edges to participate in embedding computation and exploit node-edge mutual enhancement. Finally, we present extensive experimental results to validate the improved performances of our method in terms of node classification, link prediction, and graph classification.