Jordan's classic theorem states that the center of every tree (the set of minimum eccentricity vertices) forms a complete subgraph. This property, which we refer to as the “Jordan property,” has been established for various definitions of eccentricity, the most popular being the maximum and average distances of a vertex to the others. In this note, we consider unimodal eccentricity functions, such that in every tree, the eccentricity strictly increases along every center-to-leaf path (whose second vertex is not in the center). Unimodal eccentricity implies the Jordan property. We prove that every function of distances with appropriate convexity and monotonicity is a unimodal eccentricity function. This covers many functions of distances that have been known to satisfy the Jordan property, and many others for which the Jordan property was not known prior to this work. Most of our results hold for trees with arbitrary positive weights on edges.

中文翻译：

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树中的单峰偏心率

乔丹的经典定理指出，每棵树的中心（最小偏心率顶点集）形成一个完整的子图。这个属性，我们称之为“Jordan 属性”，已经为各种离心率定义建立，最流行的是顶点到其他顶点的最大和平均距离。在本说明中，我们考虑单峰偏心率函数，使得在每棵树中，偏心率沿着每条中心到叶子的路径（其第二个顶点不在中心）严格增加。单峰偏心率意味着 Jordan 属性。我们证明了每个具有适当凸性和单调性的距离函数都是单峰偏心率函数。这涵盖了许多已知的满足 Jordan 性质的距离函数，以及在这项工作之前不知道 Jordan 性质的许多其他函数。我们的大多数结果都适用于边上具有任意正权重的树。