Extremal caterpillar-like trees relative to the Wiener index

Abstract

The Wiener index of a graph \(G\) is the sum of the distances between all vertex pairs in \(G\). A caterpillar-like tree is a tree where all its branching vertices (vertices of degree at least 3) lie on the same path. This project focuses on the class of caterpillar-like trees with given order and other restrictions such as number of leaves and maximum degree. The aim is to determine the structures of those caterpillar-like trees that reach the minimum (and maximum) Wiener index under the prescribed restrictions.