Multi-view subspace clustering methods are usually used to process high-dimensional and complex data. Most of the existing multi-view subspace clustering methods analyze and process data by mining potential graph information, with no supervision process for the representation of the potential subspace. To solve this problem, this study proposes a new multi-view subspace clustering method, namely self-supervised multi-view subspace clustering (SMSC) based on graph information. It combines spectral clustering with subspace representation to formulate a unified deep learning framework. SMSC constructs potential graph information by mining the first-order and second-order graphs of multi-view data and then uses clustering results to supervise the learning process of the common potential subspace of multi-view data. Extensive experiments on four standard datasets show that the proposed method is more effective than traditional multi-view subspace clustering methods.