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In the emerging cloud computing paradigm, data owners become increasingly motivated to outsource their complex data management systems from local sites to the commercial public cloud for great flexibility and economic savings. For the consideration of users' privacy, sensitive data have to be encrypted before outsourcing, which makes effective data utilization a very challenging task. In this paper, for the first time, we define and solve the problem of privacy-preserving query over encrypted graph-structured data in cloud computing (PPGQ), and establish a set of strict privacy requirements for such a secure cloud data utilization system to become a reality. Our work utilizes the principle of "filtering-and-verification". We prebuild a feature-based index to provide feature-related information about each encrypted data graph, and then choose the efficient inner product as the pruning tool to carry out the filtering procedure. To meet the challenge of supporting graph query without privacy breaches, we propose a secure inner product computation technique, and then improve it to achieve various privacy requirements under the known-background threat model.