Nilpotent graph structure of upper-triangular 2-by-2 matrices over finite fields

Determine whether the nilpotent graph of the Lie algebra \(\mathfrak{t}(2,\mathbb{F}_q)\) consists of \(q+1\) connected components, each of which is the complete graph \(K_{q(q-1)}\).

Background

The paper computes the nilpotent graphs of $\mathfrak{t}(2,\mathbb{F}_2)$ and $\mathfrak{t}(2,\mathbb{F}_3)$, finding respectively three and four connected components, with each component a complete graph of order $2$ and $6$. It also reports the analogous pattern for $q=4$ and $q=5$. The question asks whether this observed pattern holds for every finite field $\mathbb{F}_q$.

The paper later proves precisely this assertion in the stated theorem, so under the inclusion rule excluding questions resolved in the same paper, this question is not an unresolved open problem in the final version. However, it is explicitly posed as a question in the supplied text.