Energy finite p-harmonic functions on graphs and rough isometries

We prove that if a graph G of bounded degree has finitely many p-hyperbolic ends (1 < p < ∞) in which every bounded energy finite p-harmonic function is asymptotically constant for almost every path, then the set HBD_p(G) of all bounded energy finite p-harmonic functions on G is in one to one corresponding to R^l, where l is the number of p-hyperbolic ends of G. Furthermore, we prove that if a graph G′ is roughly isometric to G, then HBD_p(G′) is also in an one to one correspondence with R^l.