Lie algebras associated with labeled directed graphs

We construct 2-step nilpotent Lie algebras using labeled directed simple graphs and give a criterion to detect certain ideals and subalgebras by finding special subgraphs. We prove that if a label occurs only once, then reversing its orientation leads to an isomorphic algebra. As a consequence, if every edge is labeled differently, the Lie algebra depends only on the underlying undirected graph. In addition, we construct the graphs of all 2-step nilpotent Lie algebras of dimension ≤6 and compute the algebra of strata preserving derivations of the Lie algebra associated with the complete bipartite graph Km,n with two different labelings.