[Math] Is graph theory considered to be easy, or remarkably elementary?

advicediscrete mathematicsgraph theorysoft-question

I'd just like an opinion from someone more experienced. I'd like to study exciting and complex fields, and I'm wondering whether or not I should leave graph theory behind. I have somewhere between 2 and 3 semesters under my belt, but I've just begun to study Riemann Surface theory and K-Theory, both of which seem considerably more rich and complex.

Is graph theory the discrete math analogue of basic calculus? Would I be better off moving to more advanced topics?

Best Answer

It is very rare that an entire field of math can be dismissed as "easy and remarkably elementary". In all fields, there are easy questions, but they have been solved and we have moved on to the difficult questions. Basic graph theory could perhaps be compared to basic calculus, but there's plenty more to say about graph theory.

(It's conceivable that we run out of questions about a topic, because there was no depth to it. It could be argued that Euclidean geometry has reached this state. But there are certainly plenty of questions in graph theory that we still don't know the answer to.)

Now, where areas of math differ greatly is the amount of work you have to do before you get to ask a question. In graph theory, there are plenty of problems that can be asked with only the most basic of definitions - problems that you could explain to a stranger on a plane when asked "so what is it that you do?" Someone working in K-theory would have a hard time even getting to the point where a question could be formulated.

I personally find this feature attractive: I feel that a problem which is easy to state and hard to solve needs no justification to be considered interesting, whereas a problem that needs half an hour of theory to even pose is less obviously natural. And the complicated stuff is always there: that's where we get the tools to solve the simple problems. But this is a personal preference, and other mathematicians quite reasonably disagree.

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