Let A be an abelian group. A graph G = (V;E) is said to be A-barycentric-magic if there exists a labeling l : E(G) ! Anf0g such that the induced vertex set labeling l + : V (G) ! A dened by l+(v) = P uv2E(G) l(uv) is a constant map and also satises that l +(v) = deg(v)l(uvv) for all v2 V , and for some vertex uv adjacent to v. In this paper we determine all h2 N for which a given graph G is Zh-barycentric- magic and characterize Zh-barycentric-magic labeling for some graphs containing vertices of degree 2 and 3.