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?