The Zagreb indices have been introduced by Gutman and Trinajstic as M1(G) = ???V(G)(dG)(?))2 and M2(G) = ??v?E(G) dG(u)dG(?), where dG(u) denotes the degree of vertex u. We now define a new version of Zagreb indices as M*1 (G) = ??v?E(G) [?G(u)+ ?G(?)] and M*2(G)= ??v?E(G) ?G(u)?G(?), where ?G(u) is the largest distance between u and any other vertex ? of G. The goal of this paper is to further the study of these new topological index.