Abstract: Among existing clustering algorithms, the graph-Laplacian-based spectrum clustering algorithm has rigorous theoretical basis and high accuracy. However, the application of this algorithm is limited by its dependence on the prior knowledge, such as the number and the size of clusters. Based on the Jordan form of graph Laplacian, an algorithm was proposed which can obtain the prior knowledge, and perform the primary clustering based on the eigenvalues of the Jordan form. The modularity density function of clusters was defined, and an improved spectrum clustering algorithm with the help of the function and the primary clustering was proposed. The experiments were conducted on diverse datasets showing that, compared with the classic algorithms such as Fast-Newman and Girvan-Newman, the algorithm can reach a high clustering accuracy and a fast convergence rate.