I find the concept of "underlying knowledge graph" interesting. What does it mean?

I assume it means such a graph connects the topics together as "pre-requisites". To understand A you need to already understand B and C, and to understand B you need to understand D and ... etc.

But the thing about such a graph is that really it must be a tree, not just a directed graph. Why? Because there cannot be cycles in it. If to understand A you need to understand B, and to understand B you would need to understand A, you could never understand either of them. Right?

> it must be a tree, not just a directed graph

It may be a tree. But it must be a DAG (directed acyclic graph).

Heh knowledge graphs sometimes don’t feel acyclic, at least not to me anyway. Sometimes I’m stuck bouncing back and forth :)

If it's prerequisite relationship, you need to make sure that when A points to B, and B points to C, C doesn't point to A. Otherwise you're creating a loop.

it may be because the the "graph" is not accurate

Right, it must be acyclic. Which means it can be presented as a tree with some duplicate nodes. The important thing is the student must understand in which order they can try to understand the topics.

100% agreed

If to understand D you need to know both B and C, each of which requires familiarity with A, the graph is not a tree

Right, but if it is not acyclic, in which order should I try to understand them all?

If it's not acyclic then you haven't broken down the knowledge graph enough. But that's probably a waste of time, trying to come up with a perfectly ordered plan of study for all of mathematics. When you find an apparent cycle, it means the two domains are strongly interrelated and you'll be studying part of one, then the other, then the first again, repeat until you're done (whatever that means to you). No need to try and break every subject down into one-week or one-day chunks and finding a perfect ordering, just figure out the roughly course-lengthed chunks of study and start working through them, concurrently if needed as described.

Right when you read something you don't need to understand it all to understand something which may be needed to understand something else elsewhere.

But still I think it would motivate me to keep on learning if somebody could show me an accurate acyclic pre-requisites graph and tell me: "These are the thing you need to understand before you should go to the next topic. If someone could come up with the time to come up with an accurate acyclic "knowledge-graph" it would help millions of students of mathematics.

If you try hard and long enough you will understand what you're trying to understand, you will. The question is what would make that more fun and less tedious. It is about precision and not needing to learn something you don't need to learn, to understand something that you need to learn. Spend your time on learning stuff you need to learn to understand what you want to learn.

"underlying knowledge graph" is a directed acyclic graph (DAG), based on prerequisite relations among topics. So you are right that there cannot be cycles but it's not a tree either because a tree (technically) only allows one parent.