TITLE: Applications of infinite graph theory
QUESTION [33 upvotes]: Finite graph theory abounds with applications inside mathematics itself, in computer science, and engineering. Therefore, I find it naturally to do research in graph theory and I also clearly see the necessity.
Now I'm wondering about infinite graph theory. Quite a bit of research seems to be done on it as well and of course they are a natural generalization of a useful concept. But I never saw an example where we actually need them.
I understand that they come up as infinite Cayley graphs in group theory, that the automorphism groups of infinite but locally finite graphs are topological groups, that they play some role in general topology, etc. But to me it seems they are "just there" and are not essential in the sense that a theorem about them proves something about groups or topology what we couldn't have done easily without using them.
Polemically phrased my question is

Why should we care about infinite graphs?

REPLY [2 votes]: In addition to the aforementioned many applications outside graph theory, another reason for caring about infinite graphs is that they are helpful for proving results about finite graphs. The papers below give some examples.
László Babai Vertex-transitive graphs and vertex-transitive maps
https://onlinelibrary.wiley.com/doi/10.1002/jgt.3190150605
Justin Salez Sparse expanders have negative curvature
https://arxiv.org/abs/2101.08242
Martin Grohe, Pascal Schweitzer, Daniel Wiebking,
Automorphism groups of graphs of bounded Hadwiger number
https://arxiv.org/abs/2012.14300
A short proof of Fleischner’s theorem
https://www.sciencedirect.com/science/article/pii/S0012365X09003355
was found via another excursion into the infinite.
According to this wikipedia article
https://en.wikipedia.org/wiki/Halin%27s_grid_theorem,
Halin's grid theorem
"is a precursor to the work of Robertson and Seymour linking treewidth to large grid minors", which if true provides perhaps the best example of how infinite graphs help you in the study of finite ones.