I have read some part of graph theory,the question pop up in my mind, does it have application in pure mathematics? I mean like number theory,analysis,algebra,toplogy .... etc Can it be used to prove some problem in pure mathematics ? Anyone have any idea ?

3A simple google on the net would lead to possible answer to your question. For instance, you can read this [article](https://www.researchgate.net/publication/225662860_Graph_theory_to_pure_mathematics_Some_illustrative_examples) – Juniven Mar 13 '17 at 05:28

1Yes for instance the four coloring theorem revolutionized how we sometimes approach solutions. It was the first theorem to use proof by cpu – Sentinel135 Mar 13 '17 at 05:28

1Representation theory of directed graphs can answer questions like saturation, positivity, and log concavity of LR coefficients which applies to algebraic combinatorics or wider representation theory. – walkar Mar 13 '17 at 05:30

1Graphs can also be extremely useful in both group theory and constructing covering spaces in topology – Mar 13 '17 at 06:07

1The field of Dessins D'enfants is an interesting intersection of graph theory, algebra, and complex analysis. – Santana Afton Mar 13 '17 at 06:13

8Graph theory can be applied to graph theory. Please define "pure mathematics" and why graph theory isn't? – marshal craft Mar 13 '17 at 06:15

@marshalcraft I think graph theory is discrete math – Lingnoi401 Mar 13 '17 at 06:28

1@Fourier I think number theory is too but still considered pure math. In my opinion graph theory can be done in a pure math fashion. – M. Winter Mar 14 '17 at 15:29

Also see this question: https://math.stackexchange.com/questions/373845/whyshouldgraphtheorybepartoftheeducationofeverystudentofmathematic – jskattt797 Jun 04 '20 at 20:59
1 Answers
Graph Theory should be Pure Mathematics because we can study its elements in the way we would study any other geometric object in Geometry. Things get out of control when people call Inequalities Applied Mathematics or Analysis of Convex Functions Convex Analysis, but they do that... In my point of view, Inequalities and Convex Analysis are a horrendous mistake. An inequality involving Real Analysis should be studied inside of Real Analysis: if we call that inequalities, then it is a branch of Analysis, not Applied Mathematics. If we study a real convex function, that could be inside of Real Analysis. In the same way, Graph Theory should be inside of Geometry if what we are doing there is dealing with geometric objects in the traditional way: studying the shape, producing reasoning that leads to calculations, and so on.