σ-Irregularity of Trees with Prescribed Maximum Degree: A Majorization--Duality--Stability Framework · arXivDesk2608.12991Aug 13, 202622 pages, 3 figures
σ-Irregularity of Trees with Prescribed Maximum Degree: A Majorization--Duality--Stability Framework
Jasem Hamoud, Duaa Abdullah
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⩾3
. This paper develops a unified framework for this class of problems built on three pillars. A majorization-theoretic reformulation that decomposes
-extremization into a Schur-convex optimization over degree sequences and a rearrangement type optimization over tree realizations, valid for every
. We establish a duality between the maximization and minimization problems, alongside the general
closed form for
σmax(n,Δ) . A stability result showing that the optimality gap between extremal and near extremal trees is bounded independently of
and grows as
.