最佳电阻网络
Optimal Resistor Networks
摘要：
给定一个有n个顶点，m条边的图，每条边的电阻都是单位的，那么顶点对之间的平均电阻能有多小？有两种非常合理的极值结构--星形图和接近正则的图--它们之间的过渡发生在平均度为3时。然而，我们在本文中的主要目的之一是表明，对于平均度范围（包括平均度接近3），存在明显更好的构造。一个关键的想法是将这个问题与一个类似的关于有根图的问题联系起来--即“哪一个有根图使对根的平均阻力最小？”'.有根情形比无根情形分析起来简单得多，本文的主要结果之一是这两种情形是渐近等价的。
英文摘要：
Given a graph on n vertices with m edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions -- graphs like a star, and graphs which are close to regular -- with the transition between them occuring when the average degree is 3. However, one of our main aims in this paper is to show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs -- namely `which rooted graph minimises the average resistance to the root?'. The rooted case is much simpler to analyse than the unrooted, and one of the main results of this paper is that the two cases are asymptotically equivalent.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
