We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, Koolen and Markowsky (2010) [15]. Our main result shows that if the distance between points is measured by electric resistance then all points are close to being equidistant on a distance-regular graph with large valency. In particular, we show that the ratio between resistances between pairs of vertices in a distance-regular graph of diameter 3 or more is bounded by 1+6k, where k is the degree of the graph. We indicate further how this bound can be improved to 1+4k in most cases. A number of auxiliary results are also presented, including a discussion of the diameter 2 case as well as applications to random walks.
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机译：我们研究距离正则图上的电位行为，并扩展了先前论文的一些结果，Koollen and Markowsky（2010）[15]。我们的主要结果表明，如果通过电阻测量点之间的距离，则在具有大价数的距离正则图上，所有点都接近于等距。特别地，我们显示了直径为3或更大的距离正则图中一对顶点对之间的电阻比以1 + 6k为界，其中k是图的度数。我们将进一步说明在大多数情况下如何将此边界提高到1 + 4k。还提供了许多辅助结果，包括对直径2情况的讨论以及对随机游走的应用。
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