Nilpotentizers and the Nilpotent Graphs: Structural Insights into Lie Superalgebras

In this paper, we systematically investigate the nilpotentizer and nilpotent graph for a Lie superalgebra over the field of characteristic not equal to 2. First, we establish some fundamental properties of the nilpotentizer. Next, we show that the nilpotent graph is one of the isomorphic invariants of Lie superalgebras. Furthermore, we introduce the nilpotency measure which provides a quantitative assessment of nilpotency for a Lie superalgebra. Finally, we use category theory to establish connections between Lie super?algebras and their nilpotent substructures, based on the construction of the nilpotentizer.