The Quasi-Wiener and the Kirchhoff Indices 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.; Babić, D.; Trinajstić, N. J. Chem. Inf. Comput. Sci. 1993, 33, 153-154) in terms of eigenvalues of the Laplacian matrix, whereas the other is conceived (Klein, D. J.; Randić, 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 network, in which the resistance between adjacent vertices is unity. Eventually, the former quantity was named quasi-Wiener index and the latter Kirchhoff index. We now demonstrate that the quasi-Wiener and Kirchhoff indices of all graphs coincide.