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Paper IPM / M / 8727  


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(R),
is similarly defined. In this paper we investigate some
graphtheoretic properties of Γ(M_{n}(F)), where F is a
field and n\geqslant 2. Also we study the commuting graphs of
some classical groups such as GL_{n}(F) and SL_{n(}F). We show
that Γ(M_{n}(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(GL_{n}(F)) and
Γ(SL_{n}(F)). Also we show that for two fields F and E
and integers n,m \geqslant 2, if Γ(M_{n}(F)) ≅ Γ(M_{m}(E)), then n = m and F = E.
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