The European Union's Artificial Intelligence Act (AI Act) introduces comprehensive guidelines for the development and oversight of Artificial Intelligence (AI) and Machine Learning (ML) systems, with significant implications for Graph Neural Networks (GNNs). This paper addresses the unique challenges posed by the AI Act for GNNs, which operate on complex graph-structured data. The legislation's requirements for data management, data governance, robustness, human oversight, and privacy necessitate tailored strategies for GNNs. Our study explores the impact of these requirements on GNN training and proposes methods to ensure compliance. We provide an in-depth analysis of bias, robustness, explainability, and privacy in the context of GNNs, highlighting the need for fair sampling strategies and effective interpretability techniques. Our contributions fill the research gap by offering specific guidance for GNNs under the new legislative framework and identifying open questions and future research directions.