Optimal Behaviour in Extremal Bounds for $\sigma$-Irregularity

In this paper, we establishe the extremal bounds of the topological indices – Sigma index – focusing on analyzing the sharp upper bounds and the lower bounds of the Sigma index, which is known $\sigma (G)={\sum }_{uv\in E(G)}(d_G(u)-d_G(v){)}^2$. We establish precise lower and upper bounds for the Sigma index, leveraging a non-increasing degree sequence $\mathcal{D}=(d_1,d_2,\dots ,d_n)$, A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices.