Let L be a finite-dimensional Lie algebra over a field F. In This paper we introduce the nilpotent graph ΓN (L) as the graph whose vertices are the elements of L (L), where \ (L) = \x L x, y is nilpotent for all y L\, \ and where two vertices x, y are adjacent if the Lie subalgebra they generate is nilpotent. We give some characterizations of (L) and its connection with the hypercenter Z^* (L), for example, they are equal when F has characteristic zero. We prove that the nilpotentizer behaves well under direct sums, allowing a decomposition of ΓN (L) between components. The paper also investigates the structural and combinatorial properties of ΓN (L), including the conditions under which the graph is connected. We characterize the existence of strongly self-centralizing subalgebras in relation to connectivity and vertex isolation. Explicit computations are carried out for the algebra t (2, Fq), where ΓN (L) decomposes into q+1 components, each of size q (q-1), forming a (q²-q-1) -regular graph. We conclude with algorithms for constructing ΓN (L) in SageMath, and pose open problems concerning bipartiteness, regularity, and structural implications in higher dimensions over finite fields.
Towers et al. (Tue,) studied this question.