We propose a Greedy strategy to solve the problem of Graph Cut, called GGC. It starts from the state where each data sample is regarded as a cluster and dynamically merges the two clusters which reduces the value of the global objective function the most until the required number of clusters is obtained, and the monotonicity of the sequence of objective function values is proved. To reduce the computational complexity of GGC, only mergers between clusters and their neighbors are considered. Therefore, GGC has a nearly linear computational complexity with respect to the number of samples. Also, unlike other algorithms, due to the greedy strategy, the solution of the proposed algorithm is unique. In other words, its performance is not affected by randomness. We apply the proposed method to solve the problem of normalized cut which is a widely concerned graph cut problem. Extensive experiments show that better solutions can often be achieved compared to the traditional two-stage optimization algorithm (eigendecomposition + k-means), on the normalized cut problem. In addition, the performance of GGC also has advantages compared to several state-of-the-art clustering algorithms.