度量图的直径与长度
Diameter and Length of Metric Graphs
AI总结：
本文研究度量图的直径与总长度的关系，证明了带叶子数、圈秩参数的度量图总长度的上界，给出紧界及等号条件，并将结果应用于特定构造的度量图直径紧界的推导。
AI中文摘要：
度量图是一类由有限区间集合经端点成组粘合得到的度量空间，也可视为考虑每条边内部连续点集的有限边赋权图，每条边局部等距于长度为边权的区间。度量图G的直径是其所有点对间距离的最大值。本文证明：具有ℓ(G)个叶子、圈秩cyc(G)、直径diam(G)的度量图G，其总长度至多为(cyc(G)+max{1,ℓ(G)/2})·diam(G)；进一步证明该界是紧的，并刻画了等号成立的度量图。作为应用，本文对由圈或星型图经固定数量点（成对或成组）粘合得到的度量图，在特定情形下给出了直径的紧界。
英文摘要：
A metric graph is a metric space obtained from a finite collection of intervals whose endpoints are identified in groups. It can also be seen as a finite, edge-weighted graph where the continuum of points along the interior of each edge is taken into consideration, and each edge is locally isometric to an interval whose length is the edge-weight. The diameter of a metric graph $G$ is the maximum distance between all pairs of points of $G$. We show that the total length of a metric graph $G$ with $\ell(G)$ leaves, cyclomatic number $cyc(G)$, and diameter $diam(G)$ is at most $(cyc(G) + max\{1, \ell(G)/2\}) \cdot diam(G)$. Furthermore, we show that his bound is tight, and we characterize the metric graphs where equality holds. As an application, we provide tight bounds in certain cases for the diameter of metric graphs obtained from a cycle or a star by the identification of a fixed number of points (pairwise or in groups).
