The graph representation (GR) in a data space reveals the intrinsic information as well as the natural relationships of data, which is regarded as a powerful means of representation for solving the semi-supervised learning problem. To effectively learn on a pre-defined graph with both labeled data and unlabeled data, the graph convolutional network (GCN) was proposed and has attracted a lot of attention due to its high-performance graph-based feature extraction along with its low computational complexity. Nevertheless, the performance of GCNs is highly sensitive to the quality of the graph, meaning with high probability the GCNs will achieve poor performances on a badly-defined graphs. In numerous real-world semi-supervised learning problems, the graph connecting each entity in the data space implicitly exists so that there is no naturally pre-defined graph in these problems. To overcome the issues, in this paper, we apply unified graph representation (GR) techniques and graph convolutional (GC) networks in a framework that can be implemented in semi-supervised learning problems. To achieve this framework, we propose sparsity-induced graph convolutional network (SIGCN) for semi-supervised learning. SIGCN introduces the sparsity to formulate significant relationships between instances by constructing a newly-proposed L0-based graph (termed as the sparsity-induced graph), before applying graph convolution to capture the high-quality features based on this graph for label propagation. We prove and demonstrate the feasibility of the unified framework as well as effectiveness in capturing features. Extensive experiments and comparisons were performed to show the proposed SIGCN obtains a state-of-the-art performance in the semi-supervised learning problem.