# Automorphism Groups of nilpotent Lie algebras associated to certain graphs

## TL;DR

This paper studies the automorphism groups of specific 2-step nilpotent Lie algebras linked to uniform complete graphs on odd vertices, revealing their symmetry structures and subgroup relations.

## Contribution

It characterizes the automorphism groups of these Lie algebras, showing they contain dihedral groups and relate to the holomorph of cyclic groups, a novel structural insight.

## Key findings

- Automorphism group is the holomorph of Z_n.
- Contains dihedral group of order 2n as a subgroup.
- Provides explicit symmetry group descriptions for these Lie algebras.

## Abstract

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the dihedral group of order $2n$ as a subgroup.