THE QUASI-WIENER AND THE KIRCHHOFF INDEXES COINCIDE

In 1993 two novel distance-based topological indices were put forward. In the case of acyclic molecular graphs both are equal to the Wiener index, but both differ from it if the graphs contain cycles. One index is defined (Mohar, B.; Babic, D.; Trinajstic, N. J. Chem. Inf. Comput . Sci. 1993, 33, 153-154) in terms of eigenvalues of the Laplacian mat rix, whereas the other is conceived (Klein, D. J.; Randic, M. J. Math. Chem. 1993, 12, 81-95) as the sum of resistances between all pairs of vertices, assuming that the molecule corresponds to an electrical net work, in which the resistance between adjacent vertices is unity. Even tually, the former quantity was named quasi-Wiener index and the latte r Kirchhoff index. We now demonstrate that the quasi-Wiener and Kirchh off indices of all graphs coincide.