指定最大度的树的$\sigma$-不规则性：一个优化-对偶-稳定性框架
$σ$-Irregularity of Trees with Prescribed Maximum Degree: A Majorization--Duality--Stability Framework
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中文总结 AI 辅助
本文针对指定最大度$\Delta$的树的$\sigma$-不规则性问题，构建了优化-对偶-稳定性统一框架，推广了已有结果并给出$\sigma_{\max}(n,\Delta)$闭式解，证明最优性差距增长为$\Theta(\Delta^3)$。
中文摘要 AI 辅助
针对最大度为$\Delta$的树，其最大$\sigma$-不规则性已在$\Delta=4$和$\Delta=5$的情况中得到刻画，本文将该研究扩展至任意$\Delta\geqslant3$的情形。本文构建了一个统一框架，该框架基于三大支柱：一是优化理论重构，它将$\sigma$-极值问题分解为度序列上的Schur凸优化与树实现上的重排型优化，适用于所有$\Delta\geqslant3$；二是建立了最大化与最小化问题间的对偶关系，同时给出了$\sigma_{\max}(n,\Delta)$关于任意$\Delta$的闭式表达式；三是稳定性结果，表明极值树与近极值树间的最优性差距与$n$无关，增长阶为$\Theta(\Delta^3)$。
英文摘要
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 $Θ(Δ^3)$.
