Relations between Wiener, hyper-Wiener and Zagreb indices

Several inequalities are deduced, relating three previously much studied molecular structure-descriptors – the Wiener (W), hyper-Wiener (WW), and Zagreb indices (M1). Earlier reported relations between W and 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 ⩾ 3 2 n ( n - 1 ) - m - 1 2 M 1 , WW ⩾ 3 n ( n - 1 ) - 2 m - 3 2 M 1 , and WW ⩾ 2 W - 1 2 M 1 , where n and m are the number of vertices and edges of the respective molecular graph.