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Many graph construction methods for clustering cannot consider both local and global data structures in the construction of initial graph. Meanwhile, redundant features or even outliers and data with important characteristics are addressed equally in the graph optimization process. These lead to the learned representation graph may not capture the optimal structure when clustering. This paper proposes a novel model for clustering, named adaptive graph construction and low-rank representation of weighted noise (ACLWN), to overcome these problems. ACLWN is composed of an adaptive representation graph construction model named ARG, and an adaptive weighted sparse representation graph learning model named AWSG. In ARG, manifold learning and sparse representation are employed to capture the local structure of data. In AWSG, an adaptive weighted matrix is proposed to strengthen the important features and improve the robustness of the low-dimensional representation graph. Moreover, constraints such as non-negative low-rank, sparsity and distance regularization terms are imposed to capture the local and global structures of data. Comprehensive experimental results show that our method outperforms the compared state-of-the-art methods. The low-dimensional representation graph constructed by ACLWN is more suitable for clustering.