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 semisupervised learning problem. To effectively learn on a predefined 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 semisupervised learning problems, the graph connecting each entity in the data space implicitly exists so that there is no naturally predefined graph in these problems. To overcome the issues, in this article, we apply unified GR techniques and GCNs in a framework that can be implemented in semisupervised learning problems. To achieve this framework, we propose sparsity-induced graph convolutional network (SIGCN) for semisupervised learning. SIGCN introduces the sparsity to formulate significant relationships between instances by constructing a newly proposed L-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 that the proposed SIGCN obtains a state-of-the-art performance in the semisupervised learning problem.