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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?

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closed as primarily opinion-based by Somos, Paul Frost, Austin Mohr, Lee David Chung Lin, Yanior Weg Apr 28 at 7:13

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Do study graph theory. $\endgroup$ – Yves Daoust Apr 27 at 19:56
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    $\begingroup$ This question would appear to have primarily opinion based answers. I have met people who are inclined to study algebraic topics with much ease and find analysis difficult or tedious and vice-versa and there is no reason to not expect a similar phenomenon in all fields. What is considered complex or easy is subjective to an extent. $\endgroup$ – Nap D. Lover Apr 27 at 20:37
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    $\begingroup$ I'd advise you study what is most interesting to you, not necessarily what is most exciting and complex (or prestigious.) That said, the Szemerédi regularity lemma is an example of a result in graph theory that lead to a Fields medal (for the Green-Tao theorem), so I would not count out graph theory on those terms either. $\endgroup$ – Jair Taylor Apr 27 at 23:50
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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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Graph theory is incredibly relevant and there are many seemingly easy problems to which no solution has been found yet

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