Classification of $K$-forms in nilpotent Lie algebras associated to graphs

Given a simple undirected graph, one can construct from it a $c$-step nilpotent Lie algebra for every $c \geq 2$ and over any field $K$, in particular also over the real and complex numbers. These Lie algebras form an important class of examples in geometry and algebra, and it is interesting to link their properties to the defining graph. In this paper, we classify the isomorphism classes of $K$-forms in these real and complex Lie algebras for any subfield $K \subset \mathbb{C}$ from the structure of the graph. As an application, we show that the number of rational forms up to isomorphism is always one or infinite, with the former being true if and only if the group of graph automorphisms is generated by transpositions.Comment: 24 pages. The original paper was split up into two parts: one on the classification of forms in Lie algebras associated to graphs and one that will deal with the question asking which of the forms are Anosov. Both papers now also deal with the general $c$-step nilpotent case, while the original one only dealt with the 2-step nilpotent cas