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With the rapid development of 3D vision, point cloud has become an increasingly popular 3D visual media content. Due to the irregular structure, point cloud has posed novel challenges to the related research, such as compression, transmission, rendering and quality assessment. In these latest researches, point cloud quality assessment (PCQA) has attracted wide attention due to its significant role in guiding practical applications, especially in many cases where the reference point cloud is unavailable. However, current no-reference metrics which based on prevalent deep neural network have apparent disadvantages. For example, to adapt to the irregular structure of point cloud, they require preprocessing such as voxelization and projection that introduce extra distortions, and the applied grid-kernel networks, such as Convolutional Neural Networks, fail to extract effective distortion-related features. Besides, they rarely consider the various distortion patterns and the philosophy that PCQA should exhibit shifting, scaling, and rotational invariance. In this paper, we propose a novel no-reference PCQA metric named the Graph convolutional PCQA network (GPA-Net). To extract effective features for PCQA, we propose a new graph convolution kernel, i.e., GPAConv, which attentively captures the perturbation of structure and texture. Then, we propose the multi-task framework consisting of one main task (quality regression) and two auxiliary tasks (distortion type and degree predictions). Finally, we propose a coordinate normalization module to stabilize the results of GPAConv under shift, scale and rotation transformations. Experimental results on two independent databases show that GPA-Net achieves the best performance compared to the state-of-the-art no-reference PCQA metrics, even better than some full-reference metrics in some cases.