The graph convolutional networks (GCN) generalizes convolution neural networks into the graph with an arbitrary topology structure. Since the geodesic function in the null space of the graph Laplacian matrix is constant, graph Laplacian fails to preserve the local topology structure information between samples properly. GCN thus cannot learn better representative sample features by the convolution operation of the graph Laplacian based structure information and input sample information. To address this issue, this paper exploits the manifold structure information of data by the graph p-Laplacian matrix and proposes the graph p-Laplacian convolutional networks (GpLCN). As the graph p-Laplacian matrix is a generalization of the graph Laplacian matrix, GpLCN can extract more abundant sample features and improves the classification performance utilizing graph p-Laplacian to preserve the rich intrinsic data manifold structure information. Moreover, after simplifying and deducing the formula of the one-order spectral graph p-Laplacian convolution, we introduce a new layer-wise propagation rule based on the one-order approximation. Extensive experiment results on the Citeseer, Cora and Pubmed database demonstrate that our GpLCN outperforms GCN.