A graph is a structure made up of vertices and edges used to represent complex relationships between entities, while a graph stream is a continuous flow of graph updates that convey evolving relationships between entities. The massive volume and high dynamism of graph streams promote research on data structures of graph summarization, which provides a concise and approximate view of graph streams with sub-linear space and linear construction time, enabling real-time graph analytics in various domains, such as social networking, financing, and cybersecurity.In this work, we propose the Mayfly, the first neural data structure for summarizing graph streams. The Mayfly replaces handcrafted data structures with better accuracy and adaptivity.To cater to practical applications, Mayfly incorporates two offline training phases.During the larval phase, the Mayfly learns basic summarization abilities from automatically and synthetically constituted meta-tasks, and in the metamorphosis phase, it rapidly adapts to real graph streams via meta-tasks.With specific configurations of information pathways, the Mayfly enables flexible support for miscellaneous graph queries, including edge, node, and connectivity queries.Extensive empirical studies show that the Mayfly significantly outperforms its handcrafted competitors.