Our focus in this book is on using graph algorithms for analysis and feature engineering for machine learning. (More classical graph theory uses.) We do not include any content on neural networks. However, our team is extremely interested in Graph Native Learning as outlined in the Google DeepMind paper: https://blog.acolyer.org/2018/09/19/relational-inductive-biases-deep-learning-and-graph-networks/. We believe that in the future people will be running ML/DL inside graphs but this is going to take time to emerge.
1 - I don't believe you must learn graph theory to understand NN but as I mentioned above, I believe they point to some promising directions. And Graph Theory itself is just plain fun.
2- Graphs help with AI in 2 big ways today:
For graph feature engineering because relationships are often the strongest predictors of behavior. This is the focus on chapter 8 of the book.
By using Knowledge Graphs to help AI systems make better heuristic decisions by adding context.
3 - Yes! There are many many examples with sample data and code on github you can play with.
We did not include information about graph databases because it was well covered in another O'Reilly book. If you're interested in the Neo4j Graph Database, there is a book with that focus: https://neo4j.com/lp/book-graph-databases
A neural network is a graph. The very name "network" indicates that.
As to whether you need to know graph theory to understand neural nets? I don't think it's actually necessary, but it can't hurt. Everyone should know a little graph theory. You never know when it might come in handy.
Bjoke: A "Bully Joke". A Statement or action made with malicious intent - unless challenged. At which point it magically transforms into "I was just funnin'" or "What's the matter, can't take a joke?"