Complex structures on 2-step nilpotent Lie algebras arising from graphs

This work investigates the existence of complex structures on 2-step nilpotent Lie algebras arising from finite graphs. We introduce the notion of adapted complex structure, namely a complex structure that maps vertices and edges of the graph to vertices and edges, and we analyze in depth the restrictions imposed by the integrability condition. We completely characterize the graphs that admit abelian adapted complex structures, showing that they belong to a small family of graphs that we call basic. We prove that any graph endowed with an adapted complex structure $J$ contains a unique $J$-invariant basic spanning subgraph, and conversely, that every such graph can be constructed through a systematic expansion procedure starting from a basic graph. We also explore geometric and combinatorial consequences, including the existence of special Hermitian metrics as well as other graph-theoretic properties.