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- Type of Document: Article
- DOI: 10.1016/j.laa.2006.01.029
- Publisher: 2006
- Abstract:
- 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. Let D be a division ring and n ≥ 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)) ≤ 6. We conjecture that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ≤ 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. © 2006 Elsevier Inc. All rights reserved
- Keywords:
- Computational methods ; Matrix algebra ; Number theory ; Numerical methods ; Set theory ; Commuting graph ; Diameters ; Division ring ; Idempotent ; Graph theory
- Source: Linear Algebra and Its Applications ; Volume 418, Issue 1 , 2006 , Pages 161-176 ; 00243795 (ISSN)
- URL: https://www.sciencedirect.com/science/article/pii/S0024379506000590