  In this paper, we consider the formal power series whose n-th coefficient is
the number of copies of a given finite graph in the ball of radius n centred at
the identity element in the Cayley graph of a finitely generated group and call
it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the
growth function is a rational function if the group has a geodesic automatic
structure. We compute the growth function in the case where the group is
abelian and see that the denominator of the rational function is determined
from the rank of the group.
