What is the link between topology and graphs, if one exists?

Question

In spirit, topology and graph theory seems fairly similar - you have points/vertices, and a notion of "how they are connected", loosely. However, it's not obvious how these fields relate, despite topology seeming like it might be the natural generalisation of graph theory to "not-necessarily discrete graphs"; after all, it seems common for Euler's early results about graphs to be described as "the beginning of topology".

This question was inspired by seeing phrases like "the ordering relation on the real numbers induces a topology", which makes sense, but then realising that trying to do the same with the integers results in the discrete topology, which is effectively structureless with regards to the original ordering. However, you could easily represent the ordering structure using a graph (up to the direction considered positive). This seemed incongruous to me, and worth asking a question.

If there a way to subsume graphs into topology, or vice versa, or a common generalisation which subsumes them both? Please be easy on me, for I am a humble physics student.

Answer 1

Yes it does, it's called algebraic topology. For example homology tells you if a manifold is orientable. If the top dimensional integral homology group Hn is zero than it admits no orientation, if Hn is isomorphic to integers that it is orientable. To compute homology you can triangulate your space, meaning make a space which is made out of lines, vertices and triangles and is homeomorphic to your original space. Than you can easily compute all homology groups which will tell you a lot of information of how the space is connected and how many k dimensional holes it has.

Answer 2

I can think about two major ways in which graphs and topology are interlinked:

Graphs as topological spaces

You can consider a graph $G=(V,E)$ as a topological space in itself. There might be more than one way, but naturally you would identitfy $G$ with the topological space $[0,1]^E/\sim$, where you insert a copy of $[0,1]$ for each edge, and factor out a relation that identitifies the ends of these edges if they end at the same vertex.

Once you have done this, you can treat a graph like any other topological space and ask the same question, e.g. is it compact, connected, simply connected, what about its homology/homotopy, ... ?

Quantum graphs are studied to understand how solving differential equations on such topological spaces might depend on classical graph parameters (connectivity, diameter, girth, ...) or the spectrum of a graph.

Graphs embedded in other topological spaces

I think this came historically earlier. Here we try to embedd a graph in other topological space, e.g. a Euclidean space, or a manifold. If the manifold is 2-dimensional, one might ask what is the genus of the surface needed to embedd the graph injectively. E.g. you can embedd a graph injectively in the Euclidean plane if and only if it does not contain a $K_5$ or $K_{3,3}$ as (topological) minor.

Embedding a graph in a 2-manifold always induces a subdivision of that surface into smaller 2-dimensional patches. Euler's polyhedral formula is an important result here, which is generalized in algebraic topology. For such a subdivision we can define a dual graph that can help answerin map-coloring problems with purely graph theoretic methods.

For embedding graphs in 3-dimensional Euclidean space, one might aske whether one can do so without creating links. E.g. the Petersen graph cannot be embedded in 3-space without embedding two of its cycles in a linked configuration.

Answer 3

The two are deeply connected, and there is, in fact, a generalization to which both belong: Hypergraphs (see https://en.wikipedia.org/wiki/Hypergraph). An undirected hypergraph is merely a point set and a collection of subsets of that point set (typically the subsets must be nonempty, but I suspect you can accommodate allowing the empty set with the only consequence being slightly less elegant statements for some theorems). This is part of the definition of a topology (see definition by open sets here https://en.wikipedia.org/wiki/Topological_space) but a topology must do more. Hence topological spaces can be regarded as a special class of hypergraphs. Similarly, 'ordinary' graphs are a special case of hypergraphs where all of the edge subsets have exactly two elements (somewhat different, but also not unrelated in other ways too).