Premium
On ω‐commuting graphs and their diameters
Author(s) -
Raja P.,
Vaezpour S. M.
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710148
Subject(s) - mathematics , combinatorics , invertible matrix , graph , idempotence , directed graph , nilpotent , discrete mathematics , pure mathematics
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