A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems on the sum of edge lengths in trees of a fixed size: the minimum and the maximum value of the sum for specific trees, the minimum and the maximum in classes of trees (bistar trees and caterpillar trees) and finally the minimum and the maximum for any tree. We establish some foundations for research on optimality scores for spatial networks in one dimension.

中文翻译：

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树的线性排列中边长和的变化

网络科学中的一个基本问题是顶点之间的拓扑或物理距离的归一化，这需要了解非归一化距离的变化范围。在这里，我们研究了树顶点线性排列中物理距离变化的限制。特别是，我们研究了固定大小树的边长总和的各种问题：特定树的总和的最小值和最大值，树类（双星树和毛虫树）的最小值和最大值，以及最后是任何树的最小值和最大值。我们为研究一维空间网络的最优性得分奠定了一些基础。