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## “School of Mathematics”

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Paper   IPM / M / 8278
School of Mathematics
Title:   On the diameters of commuting graphs
Author(s):
 1 S. Akbari 2 A. Mohammadian (Joint with H. Radjavi and P. Raja)
Status:   Published
Journal: Linear Algebra Appl.
Vol.:  418
Year:  2006
Pages:   161-176
Supported by:  IPM
Abstract:
The commuting graph of a ring \frakR, denoted by Γ(\frakR), is a graph whose vertices are all non-central elements of \frakR and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n \geqslant 3. In this paper we investigate the diameters of Γ(Mn(D)) and determine the diameters of some induced subgraphs of Γ(Mn(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in (Mn(D)). For every field F, it is shown that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)\leqslant 6. We conjecture that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)\leqslant 5. We show that if F is an algebraically closed field or n is a prime number and Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)=4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.

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