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We prove that if a graph G of bounded degree has finitely many p-hyperbolic ends (1<p<infty)in which every bounded energy finite p-harmonic function is asymptotically constant for almost every path,then the set mathcal {HBD}_{p}(G) of all bounded energy finite p-harmonic functions on Gis 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 mathcal {HBD}_{p}(G') is also in an one to one correspondence with {bf R}^l.