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

Abstract

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