I have compiled some places where you can find graph databases. It is not an extensive list but I believe that all these websites, together, represent the main places where you can find them.
- Network repository: It has thousands of graph databases from several different domains. It is also an interactive repository, so you can visualize and explore graphs.
- Big Graph Analytics: Lists datasets and platforms available to work with graphs, including Stanfords collection (SNAP). It focuses on 'big' graph datasets.
- Awesome-Graph: Lists many graph databases and tools and frameworks used to work with graphs
- The Graphs Blog: Provides a list where you can find real and synthetic graph datasets.
- SocioPatterns: They collected data through their platform and made them available to the public. Besides the data, you can also find link to their analysis and results they had published.