In the federated clustering task, structural heterogeneity across clients inevitably impedes effective multi-source information sharing. To solve this issue, Personalized Federated Learning (PFL) has emerged as a potentially effective solution for image and text clustering. Unlike Euclidean data, graph-structured data exhibits diverse and fragile local patterns, which widely exist in real-world scenarios. Multi-graph data analysis in the federated learning setting is challenging and important, yet remains underexplored. This motivates us to propose a novel PERsonalized Federated graph-Evel Clustering Twork (PERFECT), which generates a specialized aggregation strategy for each client by uploading key model parameters and representative samples without sharing private information. Specifically, on each client, we first reconstruct privacy-preserving representative samples in a min-max optimization manner, and then upload these samples to the server for subsequent personalized parameter aggregation. On the server, we first extract graph-level embeddings from the uploaded data, and then estimate affinities among multiple learned embeddings to formulate a personalized aggregation strategy for each client. Subsequently, to help each local model better identify the cluster boundaries, we utilize clustering-wise gradient to update the key components in the personalized model parameters from the server. Extensive experimental results have demonstrated the effectiveness and superiority of PERFECT over its competitors.