刻画图中的双曲性
Characterizing Hyperbolicity in Graphs
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中文总结 AI 辅助
研究有限图双曲性刻画问题，引入与格罗莫夫δ及直径相关函数定义归一化不变量，推导出双曲平面上该函数的闭式公式，据此给出双曲平面格罗莫夫δ-双曲性最佳常数的新证明及直径缩放下标度化格罗莫夫四点条件最优常数的首个理论证明。
中文摘要 AI 辅助
格罗莫夫的δ-双曲性是度量空间树形程度的经典度量，在直径无界的空间中效果良好，但在有限图上表现不佳，主要取决于直径而非几何结构。我们引入一个将四元组的格罗莫夫δ与其直径相关联的函数，并用它定义一个归一化不变量来刻画任意有限图的双曲性，取值在零（树）和一（大晶格图）之间。我们推导出该函数在双曲平面上的闭式公式，用此公式给出双曲平面格罗莫夫δ-双曲性最佳常数的另一种证明，还给出了直径缩放下标度化格罗莫夫四点条件最优常数的首个理论证明，此前仅通过数值计算得知。
英文摘要
Gromov's delta-hyperbolicity, the classical measure of how tree-like a metric space is, works well on spaces with unbounded diameter but behaves poorly on finite graphs, where it depends primarily on diameter rather than geometry. We introduce a function relating Gromov's delta of a quadruple to its diameter and use it to define a normalized invariant that characterizes the hyperbolicity of any finite graph, taking values between zero (trees) and one (large lattice graphs). We derive a closed-form formula for this function on the hyperbolic plane. Using this formula, we give an alternate proof of the best constant of Gromov's delta-hyperbolicity for the hyperbolic plane. We also give the first theoretical proof of the optimal constant for the scaled Gromov four-point condition under diameter scaling, previously known only from numerical computations.
