Two-Step Nilpotent Lie Algebras Attached to Graphs

Consider a two-step nilpotent Lie algebra n associated with a graph as introduced in [3] endowed with an inner product for which the vertices and the edges of the graph form an orthogonal basis. We show that there exists a rank-one Einstein metric solvable extension of n if and only if the graph is positive. This generalizes the result of [6]