TITLE: Laplacian of an infinite graph and connected components
QUESTION [5 upvotes]: For a finite graph with undirected, unweighted edges, a well-known result is that the dimension of the null space of the Laplacian matrix gives the number of connected components. Does this result apply to infinite graphs as well?
The infinite graphs I'm interested are locally finite. That is, the degree of each node is finite. In my case, the number of nodes is countably infinite, there are no self-edges, and the edges are undirected.
At least according to the PDF from this course:
http://www.maths.nuigalway.ie/~rquinlan/linearalgebra/section3-1.pdf
the connected components theorem does not assume anything about finite graphs. Can someone provide a reference (a paper or text) where this is explicitly discussed?
Thank You!

REPLY [6 votes]: As Uri Bader points out, the infinite tree has an infinitely dimensional space of harmonic functions, so this is an answer to the philosophical part of the question: The way you prove that all harmonic functions on a finite graph are constant is by using a maximum principle (the value of a function at a vertex is the average over the neighbors, so if there is a vertex where the value of the function is maximal, the function must assume the same value on all the neighbors, etc. However, a function on an infinite set need not assume its supremum, which is where the argument breaks down.