Accurate and early detection of pipeline leaks and precise identification of leak severity are critical for ensuring safety and minimizing environmental and economic damage. This study proposes a two-stage fault diagnosis framework based on Graph Convolutional Networks (GCNs) for robust pipeline leak detection and leak size identification using acoustic emission (AE) signals. The proposed method transforms each AE signal into a temporal graph structure by segmenting it into equal time windows and extracting multichannel time-domain statistical features. The first stage employs a binary GCN classifier to distinguish between normal and leaking conditions. If a leak is detected, the signal is passed to a second-stage GCN that identifies the leak sizes. Experimental evaluation on a real-world pipeline dataset demonstrates the effectiveness of the proposed approach, achieving high accuracy in both leak detection and identifying leak sizes. Extensive visual analyses using t-SNE confirm the strong separability and generalization of the learned graph representations. The results validate the potential of graph-based deep learning for interpretable, accurate, and scalable pipeline health monitoring systems.