A fast and effective machine learning method, the broad learning system (BLS), has been successfully used for hyperspectral image (HSI) classification with good results. However, the original BLS cannot fully utilize the spatial information of HSI, and the linear sparse features of mapping nodes (MFs) have insufficient ability to characterize HSI. Thus, a broad graph convolutional neural network (BGCNN) is proposed for solving the aforementioned issues. In the BGCNN, the graph convolution operation is first used to capture the nonlinear spectral-spatial features, instead of only the linear sparse autoencoder in BLS. Then, the spectral-spatial features are expanded with a graph convolution operation, which further enhances the feature representation capability. Finally, the ridge regression theory is exploited to acquire the output weights. Experiments on four real HSI datasets show that our proposed BGCNN outperforms several state-of-the-art classification methods on the classification accuracy with a relatively less consumed time.