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Optimal Resistor Networks

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.