  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.
