## Abstract

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra đť”«_{G} from a simple directed graph *G* in 2005. There is a natural inner product on đť”«_{G} arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group *N* with Lie algebra đť”«_{g}. We classify singularity properties of the Lie algebra đť”«_{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 *Î“* âŠ† *N* for which the quotient *Î“* \ *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.

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