On the Extremal Zagreb Indices of $\mathbf{\textit{n}}$-Vertex Chemical Trees with Fixed Number of Segments or Branching Vertices

TL;DR

This paper characterizes chemical trees with fixed vertices, segments, or branching vertices that maximize the first and second Zagreb indices, extending previous work on their minima.

Contribution

It provides a complete characterization of extremal chemical trees with maximum Zagreb indices within specified classes, building on prior minimum index results.

Findings

Identifies trees with maximum Zagreb indices in classes with fixed segments or branching vertices.

Extends previous work on minimum Zagreb indices to maximum cases.

Offers explicit structural descriptions of extremal trees.

Abstract

Let $\mathcal{CT}_{n,k}$ and $\mathcal{CT}^*_{n,b}$ be the classes of all $n$-vertex chemical trees with $k$ segments and $b$ branching vertices, respectively, where $3\le k\le n-1$ and $1\le b< \frac{n}{2}-1$. The solution of the problem of finding trees from the class $\mathcal{CT}_{n,k}$ or $\mathcal{CT}^*_{n,b}$, with the minimum first Zagreb index or minimum second Zagreb index follows directly from the main results of [MATCH Commun. Math. Comput. Chem. 72 (2014) 825-834] or [MATCH Commun. Math. Comput. Chem. 74 (2015) 57-79]. In this paper, the chemical trees with the maximum first/second Zagreb index are characterized from each of the aforementioned graph classes.