Comaximal Graphs of finite-dimensional Lie algebras over finite fields: Triangle counts and structural invariants

Let $L$ be a finite-dimensional Lie algebra over a field $F$. The comaximal graph $\mathrm{\Gamma }(L)$ has as vertices the proper nonzero subalgebras of $L$, two of them adjacent whenever they generate $L$; its structure was previously classified for Lie algebras of dimension at most 3 over finite fields. Here we extend that work in two directions. First, we obtain explicit formulas for the number of triangles $t(\mathrm{\Gamma }(L))$ for every three-dimensional Lie algebra over ${\text{F}}_q$. Second, we extend the classification to several four-dimensional families over ${\mathbb{F}}_q$, the abelian, Heisenberg, and filiform algebras, and ${\mathfrak{g}\mathfrak{l}}_2({\text{F}}_q)$. We also relate graph-theoretic properties of $\mathrm{\Gamma }(L)$, such as completeness and the role of the Frattini subalgebra, to structural properties of $L$, including supersolvability. These results yield new combinatorial invariants for finite-dimensional Lie algebras over finite fields.