Abstract Graph convolutional neural networks (GCNs) based on spectral-domain have achieved impressive performance for semi-supervised node classification task. Recently, graph wavelet neural network (GWNN) has made a significant improvement for this task. However, GWNN is usually shallow based on a one- or two-hop neighborhood structure, making it unable to obtain sufficient global information to make it better. But, if GWNN merely stacks too many convolutional layers, it produces the phenomenon of the wavelet convolutional filters over-smoothing. To stack this challenge, we propose Multi-granularity Graph Wavelet Neural Networks (M-GWNN), a novel spectral GCNs architecture that leverages the proposed Louvain-variant algorithm and the jump connection to improve the ability of node representations for semi-supervised node classification. We first repeatedly apply the proposed Louvain-variant algorithm to aggregate nodes into supernodes to build a hierarchy of successively coarser graph, further refine the coarsened graph symmetrically back to the original by utilizing the jump connection. Moreover, during this process, multiple layers of GWNN are applied to propagate information across entire networks. The proposed M-GWNN efficiently captures node features and graph topological structures of varying granularity to obtain global information. Furthermore, M-GWNN effectively employs the jump connection to connect receptive fields of varying granularity to alleviate the speed of over-smoothing. Experiments on four benchmark datasets demonstrate the effectiveness of the proposed M-GWNN. Particularly, when only a few labeled nodes are provided on the NELL dataset, M-GWNN achieves up to an average 5.7% performance improvement compared with state-of-the-art methods.