It is important to understand the changes in the characteristics of the brain network in the state of driving fatigue and to reveal the pattern of functional connectivity between brain regions when fatigue occurs. This paper proposes a method for the detection of driving fatigue based on electroencephalogram (EEG) signals using a phase lag index graph attention network (PLI-GAT). Phase synchronization between EEG signals is a key attribute for establishing communication links among different regions of the brain, and so, the PLI is used to construct a functional brain network reflecting the relationship between EEG signals from different channels. Multi-channel EEG time-frequency features are then modeled as graph data, and the driving fatigue monitoring model is trained using a GAT. Compared with traditional graph neural networks, the GAT applies an aggregation operation to adjacent EEG channel features through the attention mechanism. This enables the adaptive assignment of different neighbor weights, which greatly improves the expressiveness of the graph neural network model. The proposed method is validated on the publicly available SEED-VIG dataset, and the accuracy of fatigue state recognition is found to reach 85.53%. The results show that the functional connectivity among different channels is significantly enhanced in the fatigue state.