Published papers about graphs can be found in dedicated journals such as the Journal of Graph Algorithms and Applications (http://jgaa.info). Papers with code also do amazing work collecting papers where the code is publicly available. The graph section (https://paperswithcode.com/area/graphs) provides a nice overview of the top current research topics regarding graphs.
However, if you can't afford to read multiple papers a week, you can still extend your knowledge about graphs and stay up to date with the latest advances by regularly checking packages dedicated to graphs. The Neo4j Graph Data Science plugin is an example of such a package.
When your graph is stored in Neo4j, the GDS is very important and already provides many interesting algorithms. However, not all of them are implemented and some applications, such as graph embedding or overlapping community detection, may require other types of algorithms. To get an overview of the existing...
