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Published online by Cambridge University Press: 06 December 2010
A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case. A well-known link with random walks motivates a conjecture about the maximum effective resistance. Arguments are given that point to the truth of the conjecture for all known distance-regular graphs.
Introduction
We shall be concerned with a graph G regarded as an electrical network in which each edge has resistance 1. A well-known result due to R.M. Foster [6] (see also [3, p.41] and [9]) asserts that if G has n vertices and m edges, the effective resistance between adjacent vertices is r1 = (n − 1)/m, provided that all edges are equivalent under the action of the automorphism group. In this paper I shall obtain formulae for ri, the effective resistance between vertices at distance i, for i ≥ 2, provided G is distance-transitive (DT). With hindsight, it will be clear that the same formulae hold if we assume only that G is distance-regular (DR). The case i = 2 was also studied by Foster [7].
Another well-known fact is that the electrical problem can be regarded as a case of solving Laplace's equation on the graph.
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