On extensions of Wiener index

Abstract

The \(n\)-th order Wiener index of a molecular graph \(G\) was put forward by E. Estrada et al. [New J. Chem. 22 (1998) 819] as \(n^W = H^{(n)} (G,x)|_{x=1}\) where \(H(G,x)\) is the Hosoya polynomial. Recently F. M. Brückler et al. [Chem. Phys. Lett. 503 (2011) 336] considered a related graph invariant, \(W^{(n)} = (1/n!)d^n (x^{n-1} H(G,x))/dx^n|_{x=1}\). For \(n=1\), both \(n^W\) and \(W^{(n)}\) reduce to the ordinary Wiener index. The aim of this paper is to obtain closed formulas for these two extensions of the Wiener index. It is proved that \(W^{(n)} =(1/n!)\sum_{k=1}^n c(n,k)W_k\) and \(n^W = \sum_{k=1}^n s(n,k)W_k\), where \(c(n,k)\), \(s(n,k)\), and \(W_k\) stand for the unsigned Stirling number of the first kind, Stirling number of the first kind, and the \(k\)-th distance moment of \(G\), respectively.