The Graph Convolutional Network(GCN) methods for skeleton-based action recognition have achieved great success due to their ability of exploiting the joint information from the graph structure of the skeleton data. Recently, as a strong and complementary modality for action recognition, the bone information from skeleton data has attracted more attention. However, most existing GCN-based methods extract the bone and joint features with two separate GCN networks, ignoring the dependencies between joints and bones. In this work, a Vertex-Edge Graph Convolutional Network(VE-GCN) is proposed to reveal the information across joints, bones and their relationships simultaneously. In addition, we learn the additional connections among joints and bones for various action samples besides the natural connections of the skeleton. Then we conduct the convolution operation on joints and their neighbors based on these additional connections. Moreover, the conditional random field (CRF) is utilized as the loss function to achieve improved performance. Experimental results on two large-scale datasets NTU RGB+D and NTU RGB+D 120 show that the proposed model outperforms state-of-the-art models.