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IPM
30
YEARS OLD

“School of Mathematics”

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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