On ω‐commuting graphs and their diameters

Abstract

Let F be a field, let ω ∈ F, and let n ⩾ 2 be a natural number. In this paper we define the ω‐commuting graph of Mn(F), denoted by Γω(Mn(F)) which is a directed graph. We prove some theorems about the strong connectivity of this graph. Also we show that the induced directed subgraphs on the set of all reducible matrices and the set of all triangularizable matrices are strongly connected graphs. Among other results we determine the diameters of the induced directed subgraphs on the sets of all non‐invertible and nilpotent matrices in Mn(F), exactly. Finally, we find good upper bounds for the diameters of some induced directed subgraphs of Γω(Mn(F)), such as the induced directed subgraphs on the set of all idempotent and diagonazable matrices in Mn(F). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim