  The minimum number of observations such that the maximum likelihood estimator
in a Gaussian graphical model exists with probability one is called the maximum
likelihood threshold of the underlying graph G. The natural algebraic
relaxation is the generic completion rank introduced by Uhler. We show that the
maximum likelihood threshold and the generic completion rank behave in the same
way under clique sums, which gives us large families of graphs on which these
invariants coincide. On the other hand we determine both invariants for
complete bipartite graphs $K_{m,n}$ and show that for some choices of m and n
the two parameters may be quite far apart. In particular, this gives the first
examples of graphs on which the maximum likelihood threshold and the generic
completion rank do not agree.
