Due to the strong propagation causality of delays between airports, this paper proposes a delay prediction model based on a deep graph neural network to study delay prediction from the perspective of an airport network. We regard airports as nodes of a graph network and use a directed graph network to construct airports' relationship. For adjacent airports, weights of edges are measured by the spherical distance between them, while the number of flight pairs between them is utilized for airports connected by flights. On this basis, a diffusion convolution kernel is constructed to capture characteristics of delay propagation between airports, and it is further integrated into the sequence-to-sequence LSTM neural network to establish a deep learning framework for delay prediction. We name this model as deep graph-embedded LSTM (DGLSTM). To verify the model's effectiveness and superiority, we utilize the historical delay data of 325 airports in the United States from 2015 to 2018 as the model training set and test set. The experimental results suggest that the proposed method is superior to the existing mainstream methods in terms of accuracy and robustness.