The Quasi-Wiener and the Kirchhoff Indices Coincide

Abstract

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.

Mean first-passage time for random walks on generalized deterministic recursive trees

“…For a tree, the Kirchhoff index is connected to the graph Laplacian eigenvalues as follows [30] Kf = n n i=2 1 λ i and as the mean first passage time for G is, by definition,…”

On the sum of powers of Laplacian eigenvalues of bipartite graphs

“…The Kirchhoff index Kf(G) of G is defined as the sum of resistance distances between all unordered pairs of vertices of G [7], [10]. As mentioned above, we have Kf(G) = ns −1 (G) .…”

“…For a connected graph G with n vertices, ns −1 (G) is equal to its Kirhhoff index [7], [13]. The Laplacian Estrada index of the graph G is defined as…”

Visualizing the Electrical Structure of Power Systems

“…Finally, note that resistance distances are closely related to random walks on graphs [69], and to the eigenvalues and eigenvectors of the graph's Laplacian, and they can be calculated by such means [70][71][72]. This is an example of spectral graph theory [54], whose application to power system problems has only recently emerged [35,61,[73][74][75][76].…”