On the Zagreb Indices as Complexity Indices

Two original Zagreb indices, denoted M1 and M2, and introduced in 1972, were symmetry-modified by summing up only degrees (SMM1) or edge-weights (SMM2) of symmetry nonequivalent vertices or edges of graphs. Their dependence on the structural features and symmetry of molecular graphs is studied. They were also compared to eight other complexity indices (RCI, TC, TC1, BT, BI, twc, wcx) on nine graphs (A, B, C, D, E, F, G, H, I) with five vertices that were earlier studied by several research groups. The TC, TC1 and Nt produce exactly the same complexity ordering of nine graphs (I>H>G>F>E>D>C>B>A). The ordering produced by M1 is different from this ordering in that it cannot discriminate E and F, and C and D. Likewise, M2 and twc produce exactly the same ordering and the latter ordering differs from the former only in the reverse order of E and F. Orderings produced by SMM1 and SMM2 differ considerably from orderings given by TC, TC1 and Nt or M2 and twc