Commuting Graphs of Matrix Algebras

ISSN： 0092-7872

Source： Communications in Algebra, Vol.36, Iss.11, 2008-11, pp. : 4020-4031

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Abstract

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(Mn(F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GLn(F) and SLn(F). We show that Γ(Mn(F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GLn(F)) and Γ(SLn(F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(Mn(F))≃Γ(Mm(E)), then n = m and |F|=|E|.