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Many relational data in our daily life are represented as graphs, making graph application an important workload. Because of the large scale of graph datasets, moving graph data to the cloud becomes a popular option. To keep the confidential and private graph secure from an untrusted cloud server, many cryptographic techniques are leveraged to hide the content of the data. However, protecting only the data content is not enough for a graph database. Because the structural information of the graph can be revealed through the database accessing track. In this work, we study the graph neural network (GNN), an important graph workload to mine information from a graph database. We find that the server is able to infer which node is processing during the edge retrieving phase and also learn its neighbor indices during GNN's aggregation phase. This leads to the leakage of the information of graph structure data. In this work, we present SPG, a structure-private graph database with SqueezePIR. Our SPG is built on top of Private Information Retrieval (PIR), which securely hides which nodes/neighbors are accessed. In addition, we propose SqueezePIR, a compression technique to overcome the computation overhead of PIR. Based on our evaluation, our SqueezePIR achieves 11.85× speedup on average with less than 2% accuracy loss when compared to the state-of-the-art FastPIR protocol.