With the rapid development of light detection and ranging (LiDAR) sensors, object detection algorithm moved quickly from traditional 2D detection to 3D detection. However, the point clouds generated by LiDAR were irregular and unstructured data, which cannot be adopted by conventional Convolutional Neural Networks. Thus, a novel Graph Convolutional Neural Network was proposed, which could better utilize the geometric relationship and topology of the data to learn features directly from point clouds for 3D object detection. First, the original LiDAR point cloud was randomly down-sampled and voxel down-sampled, and then constructed a fixed-radius neighborhood graph. Then, a novel graph convolutional neural network was designed to encode the point cloud to predict the class and shape of the object to which each vertex in the graph belongs. In order to improve the detection accuracy, we added a calibration mechanism to the network to reduce the translation error introduced by features changing in different dimensions, and also introduced an attention module to use weights to further strengthen the output vertex features. Finally, we conducted experiments on the KITTI dataset, and the experimental results showed that the method in this work could effectively detect 3D objects. Compared with many other detection algorithms, the method in this work had certain advantages in detection accuracy.