The $\omega$-commuting graph of the upper-triangular matrix algebra

In this paper, the connectivity properties and the diameter of the $\omega$-commuting graph $\Delta_{\omega}$ of the upper-triangular matrix algebra over arbitrary fields are studied.For any $n\geq 3$ and $\omega\neq 0,\pm 1$, the directed graph $\Delta_{\omega}$ is weakly connected with weak diameter $4$. As a directed graph, it has one large strongly connected component of diameter $4$ and a number of one-vertex components. In the special case of $2\times 2$ matrices, it is established that the considered graph is disconnected with components of diameter at most $2$.