# Wiener index of graphs with fixed number of pendant or cut vertices

## TL;DR

This paper characterizes the extremal graphs with fixed numbers of pendant or cut vertices that maximize or minimize the Wiener index, providing insights into the structure of such graphs.

## Contribution

It offers new characterizations of graphs with fixed pendant or cut vertices that extremize the Wiener index, filling a gap in graph theory research.

## Key findings

- Identifies graphs with maximum Wiener index given fixed pendant vertices.
- Determines graphs with minimum Wiener index given fixed cut vertices.
- Provides structural descriptions of extremal graphs.

## Abstract

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.