The development of cloud computing and the widespread application of cloud services have made outsourcing services more convenient. The need for individuals and businesses to store and manipulate the graph data they generate is growing rapidly. The unreliability and insecurity of cloud servers make outsourcing graph data a great risk of information leakage. To effectively protect data security, encrypting outsourced data is a useful method. The adjacent vertex query is a very commonly used and fundamental operation, and similarity search is a widely used and powerful tool to improve the scope and functionality of queries. After outsourcing encrypted sparse graph data to cloud servers, it becomes very inconvenient to use and manipulate the data. In this work, we present a scheme to realize the adjacent vertex query supporting similarity search on sparse graph data in cloud environment (SSAQ), which also protects the security of the information. This work uses edit distance and the searchable encryption principle to construct query index, and next implement the similar adjacent vertex query on cloud server. This work provides a formal security analysis, and also gives the experimental comparison and analysis.