Bounds on Trees with Topological Indices Among Degree Sequence

TL;DR

This paper establishes bounds and formulas relating the Albertson and Zagreb indices for trees, providing insights into their behavior based on degree sequences and contributing to topological index analysis.

Contribution

It introduces new bounds and exact formulas connecting the Albertson and Zagreb indices for trees based on degree sequences.

Findings

Derived bounds for Albertson and Zagreb indices in trees.

Established formulas linking the indices with degree sequences.

Provided insights into the relationship between topological indices and degree distributions.

Abstract

In this paper, we investigate The relationship between the Albertson index and the first Zagreb index for trees. For a tree $T=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges, we provide several bounds and exact formulas for these two topological indices, and we show that the Albertson index $\irr(T)$ and the first Zagreb index $M_1(T)$ satisfy the association \[ \operatorname{irr}(T)=d_1^2+d_n^2+(n-2)\left(\frac{\Delta + \delta}{2}\right)^2+\sum_{i=2}^{n-1} d_i+d_n - d_1-2n-2.\] Our goal of this paper is provide a topological indices, Albertson index, Sigma index among a degree sequence $\mathscr{D}=(d_1,\dots,d_n)$ where it is non-increasing and non-decreasing of tree $T$.