The Commuting Graphs of Certain Solvable Lie Algebras

TL;DR

This paper characterizes the structure of commuting graphs for solvable Lie algebras of dimension up to 4, revealing their connected components and graph properties.

Contribution

It provides a detailed description of the connected components of commuting graphs for low-dimensional solvable Lie algebras, a previously less explored area.

Findings

Connected components of commuting graphs are classified for dimension ≤ 4.

The structure of these graphs depends on the algebra's center and Lie bracket relations.

Results facilitate understanding of algebraic and graph-theoretic properties of solvable Lie algebras.

Abstract

Let $Z(\cal L)$ be the center of a Lie algebra $\cal L$ with Lie bracket $[\cdot, \cdot]$. %We then define The commuting graph of $\cal L$ is then defined by the simple undirected graph $\Gamma({\cal L})=(V_{\cal L},E_{\cal L})$ in which the vertex set is $V_{\mathcal L}=\mathcal L \setminus Z(\mathcal L)$ and the set of edges $E_{\cal L}=\left\{ \{x,y\} \mid [x,y] =0 \right\}$. The main purpose of this paper is to accurately describe the connected components of the commuting graph of solvable Lie algebras of dimension at most 4.