Sustainable management of water resources is a key challenge for the well-being and security of current and future society worldwide. In this regard, water utilities have to ensure fresh water for all users in a demand scenario stressed by climate change along with the increase in the size of cities. Dealing with anomalies, such as leakages and pipe bursts, represents one of the major issues for efficient water distribution system (WDS) operation and management. To this end, it is crucial to count on suitable methods and technologies to provide a quick, reliable, and accurate detection of such anomalies and supply disruption events. Therefore, this work proposes a novel WDS management framework based on the development of graph convolutional neural networks (GCN) models for bursts detection in WDSs. These methods rely on a WDS graph representation for a set of pressure and flow rates measures. Such a graph is used to design two GCN-based models to identify bursts. In addition, two conventional multi-layer perceptron models are used as the benchmarks to compare the graph-based methodologies. Finally, the proposed methodology is tested on a water utility network, showing the high potential of graph convolutional networks for anomaly detection on WDSs.