Let $G$ be an infinite locally finite connected graph with finitely many [ends](http://en.wikipedia.org/wiki/End_(topology)). A real-valued function $f : G \to \mathbb{R}$ is **harmonic** if 


$$f(v) = \frac{1}{d\_v} \sum\_{v \sim w} f(w)$$


where $v \sim w$ means that $v, w$ are connected by an edge. Playing around with a few examples leads me to suspect that the dimension of the space of harmonic functions on $G$ is the number of ends. (Heuristic: given a harmonic function, start with a vertex $v$ and move to a neighbor $w$ of $v$ such that $f(w) \ge f(v)$. If $f$ is nonconstant this should give a path converging to an end, and this should be possible for any end. Moreover a harmonic function should be determined by its "values at the ends.") Does anyone know if this is true and, if so, does anyone know of a reference for this fact?


(Tags are because a major application is to Cayley graphs of finitely generated groups and I would be interested in seeing how far one can push this method to prove basic facts about ends of such graphs.)



Reference request: discrete harmonic functions and ends of graphs
Life is much more complicated than that. In nice situations (for instance, if your graph is $\delta$-hyperbolic), then you can attach a more refined boundary than just the ends and (if you are lucky) solve the Dirchlet problem. A lot depends on what kinds of regularity conditions you assign to functions on the boundary at infinity.


This is by now a well-established part of geometric group theory. For instance, it plays a key role in Kleiner's recent new proof of Gromov's theorem on groups with polynomial growth. See [here](http://arxiv.org/abs/0710.4593).


One textbook reference that covers some of this information is 


Woess, W., Random Walks on Infinite Graphs and Groups, Cam- bridge Tracts in Math. 138, Cambridge Univ. Press, 2000


EDIT : By the way, since you are in Cambridge, Curt McMullen at Harvard is a good person to talk to about this kind of stuff. I learned most of what I know about the subject from a course he taught last year.


