On electric resistances for distance-regular graphs

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• 作者：Jack H. Koolen^a ; ^{koolen@postech.ac.kr} ; Greg Markowsky^b ; ^{gmarkowsky@gmail.com} ; Jongyook Park^c ; ^{jongyook@hanmail.net}
• 刊名：European Journal of Combinatorics
• 出版年：2013
• 出版时间：May, 2013
• 年：2013
• 卷：34
• 期：4
• 页码：770-786
• 全文大小：290 K

文摘

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, Koolen and Markowsky (2010) . 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 , where is the degree of the graph. We indicate further how this bound can be improved to 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.