# Generalized cut method for computing the edge-Wiener index

## TL;DR

This paper introduces a generalized method for calculating the edge-Wiener index of graphs using quotient graphs and reduction theorems, extending previous approaches and providing formulas for infinite graph families.

## Contribution

It generalizes existing methods for computing the edge-Wiener index by using weighted quotient graphs and reduction theorems, applicable to broader graph classes.

## Key findings

- Derived a closed formula for an infinite family of graphs.
- Extended methods to compute the edge-Wiener index for complex graph structures.
- Developed reduction theorems for efficient index computation.

## Abstract

The edge-Wiener index of a connected graph $G$ is defined as the Wiener index of the line graph of $G$. In this paper it is shown that the edge-Wiener index of an edge-weighted graph can be computed in terms of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of weighted quotient graphs which are defined by a partition of the edge set that is coarser than $\Theta^*$-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 so-called reduced graph. Finally, the obtained results are used to find the closed formula for the edge-Wiener index of an infinite family of graphs.