Semiregular Trees with Minimal Index

TL;DR

This paper provides a new proof that among semiregular trees of fixed size and degree, the minimal index is achieved by a caterpillar, and shows that this does not extend to trees with arbitrary degree sequences.

Contribution

It offers an alternative proof for the minimal index property of caterpillar semiregular trees and demonstrates the limitation of this property for general degree sequences.

Findings

Caterpillar semiregular trees minimize the index among fixed order and degree.

Counterexamples show the property does not hold for arbitrary degree sequences.

The new proof offers a different perspective on the minimal index problem.

Abstract

A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with index is a caterpillar. In this technical report we provide a different proof for this theorem. Furthermore, we give counter examples that show this result cannot be generalized to the class of trees with a given (non-constant) degree sequence.