Diameter of Commuting Graphs of Lie Algebras

TL;DR

This paper investigates the connectedness and diameter of commuting graphs of Lie algebras, providing methods to determine connectivity and bounds, with specific analysis on certain classes of Lie algebras.

Contribution

It introduces a process to determine connectivity and compute diameter bounds of commuting graphs for general Lie algebras and examines specific classes like low-dimensional solvable Lie algebras.

Findings

Connectedness criteria for commuting graphs

Upper bounds for diameters of these graphs

Analysis of specific Lie algebra classes

Abstract

In this paper, we study the connectedness of the commuting graph of a general Lie algebra and provide a process to determine whether the commuting graph is connected or not, as well as to compute an upper bound for its diameter. In addition, we will examine the connectedness and diameter of the commuting graphs of some remarkable classes of Lie algebras, including: (1) a class of Lie algebras with one- or two-dimensional derived algebras; and (2) a class of solvable Lie algebras over the real field of dimension up to $4$.