### Solution

I believe I have a solution. I will sketch it here. (Sorry, it's broken up into three posts because I cannot figure out how to post something more than 1000 characters...but I have seen longer solutions posted elsewhere so I assume it's okay; if not, I apologize.)

Consider the operator defined by

This is in some sense a weighting of the adjacency operator. We can then prove the result (well-known for finite graphs) that , where are the sup and inf of the spectrum of .

We note that the eigenfunctions of are simply the exponential maps .