Abstract:

The paper is devoted to  the $\omega$-commuting digraph of the full matrix algebra $M_n(\mathbb{F})$ over a field $\mathbb{F}$, where $\omega \in \mathbb{F}$  is different from $0$ and $\pm 1$. If $n=2$ it is shown that the $\omega$-commuting graph is disconnected and is a union of one strongly connected component of diameter $4$ and several strongly connected components of diameter $2$. If $n\geq 3$ and the field is algebraically closed  the $\omega$-commuting graph is strongly connected and has diameter $4$. Also it is shown that for all $n\geq 2$ the connected components and their diameters in the underlying undirected graph are the same as in the directed case.