Bounds on Trees with Topological Indices Among Degree Sequence

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 $T$.