  We introduce a combinatorial method to construct indefinite Ricci-flat
metrics on nice nilpotent Lie groups.
  We prove that every nilpotent Lie group of dimension $\leq6$, every nice
nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group
attached to a graph admits such a metric. We construct infinite families of
Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie
groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these
metrics are shown not to be flat.
