Rolling bearings are essential parts of rotating equipment. Due to their unique operating environment, bearings are vulnerable to failure. Graph neural network (GNN) provides an effective way of mining relationships between data samples. However, various existing GNN models suffer from issues like poor graph-structured data quality and high computational consumption. Moreover, the available fault samples are typically insufficient in real practice. Therefore, an improved graph convolutional network (GCN) is proposed for bearing fault diagnosis with limited labeled data. This method consists of two steps: graph structure data acquisition and improved graph convolution network building. Defining edge failure thresholds simplifies the generated weighted graph-structured data, thereby enhancing data quality and reducing training computation costs. Improvements to standard GCNs can effectively aggregate data features of different receptive field sizes without noticeably raising the computational complexity of the model. Experiments with limited labeled data are conducted on two public datasets and an actual experimental platform dataset to verify the superiority of the proposed method. In addition, experiments on imbalanced datasets also fully demonstrate the robustness of the proposed method.