Some Results on The Non-Commuting Graph of Finite-Dimensional Lie Algebras

Let [Formula: see text] be a non-abelian finite-dimensional Lie algebra and [Formula: see text] be its centre. Then the non-commuting graph of the Lie algebra [Formula: see text] is defined as an undirected simple graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we consider all Lie algebras [Formula: see text] with [Formula: see text] and draw the non-commuting graph of such Lie algebras. Also, we determine some graph properties of these kinds of graphs such as maximum and minimum degrees, planarity, outer planarity, independence, domination, clique and chromatic numbers. Moreover, the structures of non-commuting graphs of a Lie algebra [Formula: see text] with a given minimum degree are explained. Some upper and lower bounds for invariant parameters are stated, and the structure of regular non-commuting graphs of Lie algebras is determined at the end.