The comaximal graph of a finite-dimensional Lie algebra

In this paper, we introduce the comaximal graph [Formula: see text] of a finite-dimensional Lie algebra L, whose vertices are the nontrivial proper Lie subalgebras of L over a field [Formula: see text], and two vertices A and B are adjacent if and only if [Formula: see text]. We establish general structural properties, including a characterization of isolated vertices via the Frattini subalgebra and a criterion for completeness in terms of [Formula: see text]-algebras. We classify [Formula: see text] for all Lie algebras of dimension at most three over a finite field [Formula: see text], providing an explicit description in each case. The resulting graphs exhibit a rich range of behaviors, depending on the structure of the derived algebra and the action of ad x. For [Formula: see text], we determine several graph invariants, including the degree sequence, clique number, chromatic number, domination number, diameter, and radius, and show that [Formula: see text] is connected and non-planar. The graph contains a large clique formed by the nonsplit semisimple lines together with the Borel subalgebras, while the nilpotent and split semisimple lines have a more restricted adjacency structure governed by their containment in Borel subalgebras.