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On ω-commuting graphs and their diameters

✍ Scribed by P. Raja; S. M. Vaezpour

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
127 KB
Volume
284
Category
Article
ISSN
0025-584X

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✦ Synopsis

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 M~n~(F), denoted by Γ~ω~(M~n~(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 M~n~(F), exactly. Finally, we find good upper bounds for the diameters of some induced directed subgraphs of Γ~ω~(M~n~(F)), such as the induced directed subgraphs on the set of all idempotent and diagonazable matrices in M~n~(F). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

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