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Automatic labelling of anatomical structures, such as coronary arteries, is critical for diagnosis, yet existing (non-deep learning) methods are limited by a reliance on prior topological knowledge of the expected tree-like structures. As the structure such vascular systems is often difficult to conceptualize, graph-based representations have become popular due to their ability to capture the geometric and topological properties of the morphology in an orientation-independent and abstract manner. However, graph-based learning for automated labeling of tree-like anatomical structures has received limited attention in the literature. The majority of prior studies have limitations in the entity graph construction, are dependent on topological structures, and have limited accuracy due to the anatomical variability between subjects. In this paper, we propose an intuitive graph representation method, well suited to use with 3D coordinate data obtained from angiography scans. We subsequently seek to analyze subject-specific graphs using geometric deep learning. The proposed models leverage expert annotated labels from 141 patients to learn representations of each coronary segment, while capturing the effects of anatomical variability within the training data. We investigate different variants of so-called message passing neural networks. Through extensive evaluations, our pipeline achieves a promising weighted F1-score of 0.805 for labeling coronary artery (13 classes) for a five-fold cross-validation. Considering the ability of graph models in dealing with irregular data, and their scalability for data segmentation, this work highlights the potential of such methods to provide quantitative evidence to support the decisions of medical experts.