Graphs and metric 2-step nilpotent Lie algebras

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra G from a simple directed graph G in 2005. There is a natural inner product on G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra g. We classify singularity properties of the Lie algebra g in terms of the graph G. A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property