Federated learning enables collaborative training of multiple participants and aggregation to generate global models while protecting data privacy. However, this approach is challenged by the substantial communication overhead imposed by the transmission of model parameters from clients to the fusion server in the process of obtaining a global model. The accuracy of the global model may decrease on clients due to concept drift between client data. To resolve these problems, we propose a communication efficient Fuzzy Clustered Graph Federated Learning (FCGFL) approach, where a graph federated learning method based on a personalized decoupling idea is designed, with clients uploading shared layer parameters to reduce the number of transmitted model parameters to improve communication efficiency. A fuzzy clustered federated learning method based on multi-step matrix optimization is designed in FCGFL to improve the model accuracy of federated learning by optimizing the affiliation matrix of fuzzy clustering, and to enhance the convergence speed to reduce the number of communication rounds. FCGFL is evaluated on public datasets and also industrial defect scenarios, and the results show that FCGFL outperforms the existing federated learning algorithms, with a reduction in communication time per round of 5%, a reduction in the average number of rounds for convergence of about 40%, and an improvement in average model accuracy of 2%.