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?