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?
Best Answer
The first book on graph theory was König's Theorie der endlichen und unendlichen Graphen (Theory of finite and infinite graphs) of 1936. Thus infinite graphs were part of graph theory from the very beginning. König's most important result on infinite graphs was the so-called König infinity lemma, which states that in an infinite, finitely-branching, tree there is an infinite branch. This lemma encapsulates many arguments -- from the Bolzano-Weierstrass theorem, to the completeness theorem of logic, to the proof of various Ramsey theorems -- in graph-theoretic form. König himself used it to prove that the infinite form of van der Waerden's theorem on arithmetic progressions implies the finite version, and Erdos and Szekeres (who were students of König) took up the idea in their pioneering 1935 paper on Ramsey theory.
As other commentators have mentioned, infinite graphs are also important as group diagrams in combinatorial group theory and low-dimensional topology.