We present a novel graph-based approach for labeling the anatomical branches of a given airway tree segmentation. The proposed method formulates airway labeling as a branch classification problem in the airway tree graph, where branch features are extracted using convolutional neural networks (CNN) and enriched using graph neural networks. Our graph neural network is structure-aware by having each node aggregate information from its local neighbors and position-aware by encoding node positions in the graph. We evaluated the proposed method on 220 airway trees from subjects with various severity stages of Chronic Obstructive Pulmonary Disease (COPD). The results demonstrate that our approach is computationally efficient and significantly improves branch classification performance than the baseline method. The overall average accuracy of our method reaches 91.18\% for labeling all 18 segmental airway branches, compared to 83.83\% obtained by the standard CNN method. We published our source code at https://github.com/DIAGNijmegen/spgnn. The proposed algorithm is also publicly available at https://grand-challenge.org/algorithms/airway-anatomical-labeling/.