Classical invariants of the non-nilpotent graph

Investigate the classical graph-theoretic invariants of the non-nilpotent graph associated with a finite-dimensional Lie algebra, whose vertices are \(L\setminus nil(L)\) and whose edges join pairs generating a non-nilpotent subalgebra.

Background

The paper introduces the non-nilpotent graph as the complement of the nilpotent graph on the same vertex set. It observes that this complement graph is connected and illustrates it for $\mathfrak{t}(2,\mathbb{F}_2)$.

The authors leave unresolved the broader structural and combinatorial analysis of this graph, asking what can be said about its classical invariants, such as connectivity-related parameters, degree data, clique and independence numbers, diameter, and spectral quantities.