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Paper   IPM / M / 8727
School of Mathematics
  Title:   Commuting graphs of matrix algebras
  Author(s):  S. Akbari (Joint with H. Bidkhori and A. Mohammadian)
  Status:   Published
  Journal: Comm. Algebra
  Year:  2008
  Pages:   4020-4031
  Supported by:  IPM
The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central 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(R), is similarly defined. In this paper we investigate some graph-theoretic properties of Γ(Mn(F)), where F is a field and n\geqslant 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 \geqslant 2, if Γ(Mn(F)) ≅ Γ(Mm(E)), then n = m and |F| = |E|.

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