On the diameters of commuting graphs |
| |
Authors: | S Akbari A Mohammadian H Radjavi |
| |
Institution: | a Institute for Studies in Theoretical Physics and Mathematics, P. O. Box 19395-5746, Tehran, Iran b Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365-9415, Tehran, Iran c Department of Pure Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 |
| |
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. |
| |
Keywords: | 05C50 15A27 15A33 16P10 |
本文献已被 ScienceDirect 等数据库收录! |
|