Spectral clustering is an empirically successful approach to separating a dataset into some groups with possibly complex shapes based on pairwise affinity. Identifying the number of clusters automatically is still an open issue, although many heuristics have been proposed. In this paper, imposing sparsity on the eigenvectors of graph Laplacian is proposed to attain reasonable approximations of the so-called cluster-indicator-vectors, from which the clusters as well as the cluster number are identified. The proposed algorithm enjoys low computational complexity as it only computes a relevant subset of eigenvectors. It also enjoys better clustering quality than the existing methods, as shown by simulations using nine real datasets.