  Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation
in terms of Cayley graphs, we prove the following main result : There is a
universal constant e > 0 such that, if G is a connected, regular graph on n
vertices, then either every pair of vertices can be connected by a path of
length at most 3, or the number of pairs of such vertices is at least 1+e times
the number of edges in G. We discuss a range of further questions to which this
result gives rise.
