Graph unlearning, which deletes graph elements such as nodes and edges from trained graph neural networks (GNNs), is crucial for real-world applications where graph data may contain outdated, inaccurate, or privacy-sensitive information. However, existing methods often suffer from (1) incomplete or over unlearning due to neglecting the distinct objectives of different unlearning tasks, and (2) inaccurate identification of neighbors affected by deleted elements across various GNN architectures. To address these limitations, we propose AGU, a novel Adaptive Graph Unlearning framework that flexibly adapts to diverse unlearning tasks and GNN architectures. AGU ensures the complete forgetting of deleted elements while preserving the integrity of the remaining graph. It also accurately identifies affected neighbors for each GNN architecture and prioritizes important ones to enhance unlearning performance. Extensive experiments on seven real-world graphs demonstrate that AGU outperforms existing methods in terms of effectiveness, efficiency, and unlearning capability.