What’s the relation between topology and graph theory

I read the Wikipedia articles for both topology, graph theory (plus topological graph theory). Does topology encompass also graph theory? Or topology is only about studying shapes while graph theory is about relations and the two meet in topological graph theory?

Answer

There are (at least) two ways to answer this question. In the strict definitional sense, you can probably get all of graph theory expressed in the language of topology. If you’re really sneaky you can probably do it the other way, too, so you could probably have a good time claiming that “all of graph theory is just part of topology”, and likewise “all of topology is just part of graph theory”.

However, more importantly I think, the flavour of the two fields are typically quite different. By this I mean, if you happen upon a mathematician these days that considers herself a topologist, chances are she works either on something geometric, or very algebraic, and either way something pretty abstract. On the other hand, if you happen upon a mathematician that considers herself a graph theorist, chances are she works on some pretty concrete objects, possibly with more obvious direct connections to real world applications.

The exceptions to that paragraph are numerous, but sort of prove the rule. There are plenty of topologists that work on very concrete objects, and plenty of graph theorists whose work is very abstract or use tools from algebraic geometry, number theory, etc.

Nonetheless, it’s unlikely the two will share much overlap in their mathematical interests, and if they do, you would probably call their shared interest topological graph theory, or combinatorial topology. So, I think your last statement is closest to the truth, although I’d say topology is much more than studying shapes, and graph theory is much more than studying relations ( or at least doesn’t reel like you’re studying relations when you’re doing it ). Examples are probably the best way to demonstrate that, but then this answer will get way too long, and you’re better off just reading/doing some of each to get a feeling for what each field is about.

Attribution
Source : Link , Question Author : Francesco , Answer Author : Callus – Reinstate Monica

Leave a Comment