On the Edge Wiener index

Abstract

Let G be a simple connected graph. The Wiener index of G is the sum of all distances between vertices of G. Whereas, the edge Wiener index of G is defined as the sum of distances between all pairs of edges of G where the distance between the edges f and in E(G) is defined as the distance between the vertices f and in the line graph of G. In this paper we will describe a new method for calculating the edge Wiener index. Then find this index for the triangular graphs. Also, we obtain an explicit formula for the Wiener index of the Cartesian product of two graphs using the group automorphisms of graphs.

Edge Distance‐based Topological Indices of Strength‐weighted Graphs and their Application to Coronoid Systems, Carbon Nanocones and SiO2 Nanostructures

“…In the recent years, mathematical techniques for the computation of the edge‐Wiener index have been considered by a number of researchers, see the papers and a survey . Motivated by the vertex and edge versions of the Wiener indices, the vertex‐edge‐Wiener index was introduced in the paper and it is defined as the sum of distances between all pairs of G consisting of a vertex and an edge.…”