Abstract
In this paper, we consider an asymptotic degree sequence denotedby \(\mathcal{D}=(d_{1},d_{2},\dots,d_{n})\) , where \(d_{n}\geqslant\dots\geqslant d_{1}\) . The examination of topological indices ontrees provides a general overview through bounds that identify themaximum and minimum number of edges incident to each vertex in thegraph. We study the Albertson index, defined as \(\sum_{uv\in E(G)}|d_{u}(G)-d_{v}(G)|\) , and the Sigma index \(\sigma(G)\) for atree \(T\) with degree sequence \(\mathcal{D}\) . Using the firstZagreb index \(M_{1}(T)\) , we show that for a degree sequence of order \(n=4\) , the irregularity of a tree \(T\) is given by \(\textrm{irr}(T)=M_{1}(T)^{2}-2\sqrt{M_{1}(T)}+\sum_{i=1}^{3}|d_{i}-d_{i+1}|-(d_{2}+d_{3})-1\) .
