Generalized cut method for computing the edge-Wiener index

Abstract

The Wiener index is a well known topological descriptor defined as the sum of distances between all the pairs of vertices in a connected graph. The edge-Wiener index of a graph G is then defined as the Wiener index of the line graph of G. In this paper, we show that the edge-Wiener index of an edge-weighted graph can be computed in terms of the three Wiener indices of weighted quotient graphs obtained from a partition of the edge set that is coarser than the Θ∗-partition. Thus, already known analogous methods for computing the edge-Wiener index of benzenoid systems and phenylenes are greatly generalized. Moreover, reduction theorems are developed for the edge-Wiener index and the vertex–edge-Wiener index since they can be applied in order to compute a corresponding index of a (quotient) graph from the reduced graph. Finally, the obtained results are used to find the closed formula for the edge-Wiener index of an infinite family of graphs.