The accuracy of point cloud registration (PCR) algorithm based on feature matching is often affected by noise and repeated local structures in point clouds. In this paper,an L1,P sparse constraint model-based point cloud registration algorithm is proposed to address these issues. First, geometric or learned point cloud feature descriptors are extracted to construct compatibility graphs using topological relationships, and the spectral matching method is employed to select seed points with higher confidence scores. Second, the discrete constraint of graph matching is relaxed into L1,P sparse constraints to obtain optimal matching results for the fully connected attribute-weighted graph model constructed from the seed points. Finally, a PCR method based on maximal clique hypothesis testing is realized by searching for maximal cliques within the matching results and scoring them, and then selecting the highest-scoring transformation pose as the precise registration pose. Experimental results demonstrate that the proposed algorithm achieves a maximum accuracy improvement of 72.12% and 75.86% on indoor and outdoor point cloud datasets, respectively, compared to typical registration algorithms. In addition, under different parameters settings and Gaussian noise interference, the proposed algorithm demonstrates superior registration accuracy and robustness compared to other registration algorithms.