A graph theoretical determination of solvable complete rigid Lie algebras

We describe a class of nilpotent Lie algebras completely determined by their associated weight graph. These algebras also present two important structural properties: to admit naturally a symplectic form and to be isomorphic to the nilradical of a solvable complete rigid Lie algebra. These solvable algebras are proved to constitute a class of algebras where a symplectic form cannot exist. Finally we analyze the product by generators of the preceding algebras, and show that this operator preserves the property of being the maximal nilpotent ideal of a solvable rigid Lie algebra.