$\sigma$-Irregularity of Trees with Prescribed Maximum Degree: A Majorization--Duality--Stability Framework

Trees of maximum $\sigma$-irregularity subject to a prescribed maximum degree $\mathrm{\Delta }$ have been characterized for $\mathrm{\Delta }=4$ and $\mathrm{\Delta }=5$, with the extension to arbitrary $\mathrm{\Delta }⩾3$. This paper develops a unified framework for this class of problems built on three pillars. A majorization-theoretic reformulation that decomposes $\sigma$-extremization into a Schur-convex optimization over degree sequences and a rearrangement type optimization over tree realizations, valid for every $\mathrm{\Delta }⩾3$. We establish a duality between the maximization and minimization problems, alongside the general $\mathrm{\Delta }$ closed form for ${\sigma }_{\max }(n,\mathrm{\Delta })$. A stability result showing that the optimality gap between extremal and near extremal trees is bounded independently of $n$ and grows as $\mathrm{\Theta }({\mathrm{\Delta }}^3)$.