Minimizing Degree-based Topological Indices for Trees with Given Number of Pendent Vertices + Erratum

TL;DR

This paper establishes sharp lower bounds for Zagreb indices in trees with a given number of pendent vertices, identifies optimal tree structures, and corrects a previous theorem with an erratum.

Contribution

It derives optimal tree configurations minimizing Zagreb indices and generalizes the technique to other degree-based indices, also correcting a previous theoretical error.

Findings

Minimizers for $M_1$ and $M_2$ are identified.

The technique applies to weighted and other degree-based indices.

An erratum corrects the bounds for $M_2$ when $n ge 8$.

Abstract

We derive sharp lower bounds for the first and the second Zagreb indices ($M_1$ and $M_2$ respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. $M_1$ is minimized by a tree with all internal vertices having degree 4, while $M_2$ is minimized by a tree where each "stem" vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the rest internal vertices are incident to 3 other internal vertices. The technique is shown to generalize to the weighted first Zagreb index, the zeroth order general Randi\'{c} index, as long as to many other degree-based indices. Later the erratum was added: Theorem 3 says that the second Zagreb index $M_2$ cannot be less than $11n-27$ for a tree with $n\ge 8$ pendent vertices. Yet the tree exists with $n=8$ vertices (the two-sided broom) violating this inequality. The reason is that the proof of Theorem 3 relays on a tacit assumption that an index-minimizing tree contains no vertices of degree 2. This assumption appears to be invalid in general. In this erratum we show that the inequality $M_2 \ge 11n-27$ still holds for trees with $n\ge 9$ vertices and provide the valid proof of the (corrected) Theorem 3.