A short note on 2-step nilpotent Lie algebras associated with graphs

In this note we consider 2-step nilpotent Lie algebras associated with graphs. We prove that 2-step nilpotent Lie algebras $\text{n}$ and ${\text{n}}^{\mathrm{\prime }}$ associated with graphs $(S,E)$ and $(S^{\mathrm{\prime }},E^{\mathrm{\prime }})$ respectively are isomorphic if and only if $(S,E)$ and $(S^{\mathrm{\prime }},E^{\mathrm{\prime }})$ are isomorphic.