Currently, deep learning (DL) algorithms have achieved great success in many applications including computer vision and natural language processing. Many different kinds of DL models have been reported, such as DeepWalk, LINE, diffusionconvolutional neural networks, graph convolutional networks (GCN), and so on. The GCN algorithm is a variant of convolutional neural network and achieves significant superiority by using a one-order localised spectral graph filter. However, only a one-order polynomial in the Laplacian of GCN has been approximated and implemented, which ignores undirect neighbour structure information. The lack of rich structure information reduces the performance of the neural networks in the graph structure data. In this study, the authors deduce and simplify the formula of two-order spectral graph convolutions to preserve rich local information. Furthermore, they build a layerwise GCN based on this two-order approximation, i.e. two-order GCN (TGCN) for semi-supervised classification. With the two-order polynomial in the Laplacian, the proposed TGCN model can assimilate abundant localised structure information of graph data and then boosts the classification significantly. To evaluate the proposed solution, extensive experiments are conducted on several popular datasets including the Citeseer, Cora, and PubMed dataset. Experimental results demonstrate that the proposed TGCN outperforms the state-of-art methods.