Several inequalities are deduced, relating three previously much studied molecular structure-descriptors - the Wiener (W), hyper-Wiener (WW), and Zagreb indices (M-1). Earlier reported relations between Wand WW for trees are now extended so as to apply to all molecular graphs (except those containing triangles and quadrangles), and the earlier estimates for WW of trees are improved. The main relations obtained are: W greater than or equal to (3)/(2)n(n - 1) - m - (1)/M-2(1), WW greater than or equal to 3n(n - 1) - 2m - (3)/M-2(1), and WW greater than or equal to 2W - (1)/M-2(1), where n and m are the number of vertices and edges of the respective n olecular graph. 2 (C) 2004 Elsevier B.V. All rights reserved.