I learned a little about them at school (long time ago). Relational not yet completely captured the world, but it was close, so the graph database got a superficial mention. IIRC they were pretty much dead at the time. I'm not sure how informative it really will be, but I will post it there if it helps someone.
Basically, if what I remember is true, the graph database is essentially a graph. You extract data (nodes) from the graph, and then, to find the relevant information, you move links (edges) to the associated data in the graph structure.
Apart from the obvious case where your data is graphical, and it may be faster / more natural to use, I cannot recall any advantages. I don’t remember any of the shortcomings, but I would suggest that it might not work well with those things that relational databases do (for example, using large sets of tuples).
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