Bounds on Trees with Topological Indices Among Degree Sequence

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 $n=\mathrm{\mid }V\mathrm{\mid }$ vertices and $m=\mathrm{\mid }E\mathrm{\mid }$ edges, we provide several bounds and exact formulas for these two topological indices, and we show that the Albertson index $\text{irr}(T)$ and the first Zagreb index $M_1(T)$ satisfy the association $$\mathrm{irr}(T)=d_1^2+d_n^2+(n-2){\left(\frac{\mathrm{\Delta }+\delta }{2}\right)}^2+\sum _{i=2}^{n-1}{d}_i+d_n-d_1-2n-2.$$ Our goal of this paper is provide a topological indices, Albertson index, Sigma index among a degree sequence $\mathcal{D}=(d_1,\dots ,d_n)$ where it is non-increasing and non-decreasing of tree $T$.