Graphs and and two-step nilpotent Lie algebras

TL;DR

This paper establishes a precise correspondence between finite simple graphs and two-step nilpotent Lie algebras, proving that graph isomorphism is equivalent to Lie algebra isomorphism in this context.

Contribution

The paper proves that two graphs produce isomorphic two-step nilpotent Lie algebras if and only if the graphs themselves are isomorphic, clarifying the relationship between graph theory and Lie algebra structures.

Findings

Lie algebras associated with isomorphic graphs are Lie isomorphic

Graph isomorphism corresponds exactly to Lie algebra isomorphism in this construction

Provides a characterization linking graph isomorphism to Lie algebra isomorphism

Abstract

We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the graphs from which they arise are isomorphic.