  Inspired by the work of Backelin on non-commutative correspondences to
Macaulay's theorem of the growth of the Hilbert series of affine algebras, we
study embedding dimension dependant versions of his degree 2 to degree 3
result. In graph-theoretical terms, we study the following question: what is
the maximal number of directed walks of length 2 in a digraph with (k) edges
and (n) vertices? The problem can also be formulated as follows: maximize (<
\lambda, \lambda^T >) when (\lambda) is a partition of (k), contained in an (n
\times n) box. We show that for mild restrictions on (n), optimal digraphs are
the ``stars of saturated stars''.
