  This note gives three short results concerning the elliptic Harnack
inequality (EHI), in the context of random walks on graphs. The first is that
the EHI implies polynomial growth of the number of points in balls, and the
second that the EHI is equivalent to an annulus type Harnack inequality for
Green's functions. The third result uses the lamplighter group to give a
counterexample concerning the relation of coupling with the EHI.
