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On the diameters of commuting graphs

✍ Scribed by S. Akbari; A. Mohammadian; H. Radjavi; P. Raja

Publisher: Elsevier Science
Year: 2006
Tongue: English
Weight: 200 KB
Volume: 418
Category: Article
ISSN: 0024-3795
DOI: 10.1016/j.laa.2006.01.029

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✦ Synopsis

The commuting graph of a ring R, denoted by (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 (M n (D)) and determine the diameters of some induced subgraphs of (M n (D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M n (D). For every field F , it is shown that if (M n (F )) is a connected graph, then diam (M n (F )) 6. We conjecture that if (M n (F )) is a connected graph, then diam (M n (F )) 5. We show that if F is an algebraically closed field or n is a prime number and (M n (F )) is a connected graph, then diam (M n (F )) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.

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