  It is known that complete graphs and complete multipartite graphs have
modularity zero. We show that the least number of edges we may delete from the
complete graph $K_n$ to obtain a graph with non-zero modularity is $\lfloor
n/2\rfloor +1$. Similarly we determine the least number of edges we may delete
from or add to a complete bipartite graph to reach non-zero modularity. We give
some corresponding results for complete multipartite graphs, and a short proof
that complete multipartite graphs have modularity zero.
  We also analyse the modularity of very dense random graphs, and in particular
we find that there is a transition to modularity zero when the average degree
of the complementary graph drops below 1.
