AI 中文总结
本文为图建立Krein-Milman型定理和Ulam型稳定性结果，引入顶点加权凸图的定义，揭示序贯凸性与完美二叉树的关系，通过最小权重扰动使近似凸性精确，研究结构特征、凸次优项与三明治型结果，还探究了树的极值问题。
AI 中文摘要
本文的主要目标是为图建立Krein-Milman型定理和Ulam型稳定性结果。为确立这些结果，受序贯凸性概念启发，我们引入了顶点加权凸图的若干有意义定义。我们还揭示了序贯凸性与完美二叉树这两个离散结构之间的密切关系。我们证明，若一个图近似满足某一凸性性质，则通过对其顶点分配的权重进行最小扰动，可使该性质变得精确。此外，我们研究了若干结构特征，为加权图构建凸次优项，并推导了三明治型结果。研究特别强调树，还对极值问题进行了探究。
英文摘要
The main objective of this paper is to develop Krein-Milman-type theorems and Ulam-type stability results for graphs. To establish these results, we introduce several meaningful definitions of vertex-weighted convex graphs inspired by the concept of sequential convexity. We also present a close relationship between the two discrete structures, namely sequential convexity and perfect binary trees. We show that if a graph satisfies a certain convexity property approximately, then this property can be made exact by minimally perturbing the weights assigned to its vertices. Furthermore, we study several structural characterisations, formulate convex minorants for weighted graphs, and derive sandwich-type results. Special emphasis is placed on trees, and an investigation of extremal value problems is also carried out
