Point cloud registration is a fundamental task in many applications such as localization, mapping, tracking, and reconstruction. Successful registration relies on extracting robust and discriminative geometric features. Though existing learning-based methods require high computing capacity for processing a large number of raw points at the same time, computational capacity limitation is not an issue thanks to powerful parallel computing process using GPU. In this paper, we introduce a framework that efficiently and economically extracts dense features using a graph attention network for point cloud matching and registration (DFGAT). The detector of the DFGAT is responsible for finding highly reliable key points in large raw data sets. The descriptor of the DFGAT takes these keypoints combined with their neighbors to extract invariant density features in preparation for the matching. The graph attention network (GAT) uses the attention mechanism that enriches the relationships between point clouds. Finally, we consider this as an optimal transport problem and use the Sinkhorn algorithm to find positive and negative matches. We perform thorough tests on the KITTI dataset and evaluate the effectiveness of this approach. The results show that this method with the efficiently compact keypoint selection and description can achieve the best performance matching metrics and reach the highest success ratio of 99.88% registration in comparison with other state-of-the-art approaches.