# On the diameter of the commuting graph of the full matrix ring over the real numbers

Document Type : Research Paper

Authors

1 Departamento de Matemáticas‎, ‎Universidad de Oviedo‎. ‎Avenida Calvo Sotelo s/n‎, ‎33007 Oviedo‎, ‎Spain.

2 Centro Universitario de la Defensa de Zaragoza‎. ‎Carretera de Huesca s/n‎, ‎50090 Zaragoza‎, ‎Spain.

Abstract

‎In a recent paper C‎. ‎Miguel proved that the diameter of the commuting graph of the matrix ring $\mathrm{M}_n(\mathbb{R})$ is equal to $4$ if either $n=3$ or $n\geq5$‎. ‎But the case $n=4$ remained open‎, ‎since the diameter could be $4$ or $5$‎. ‎In this work we close the problem showing that also in this case the diameter is $4$.

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#### References

S. Akbari, M. Ghandehari, M. Hadian and A. Mohammadian, On commuting graphs of semisimple rings, Linear Algebra Appl. 390 (2004) 345--355.
S. Akbari, D. Kiani and F. Ramezani, Commuting graphs of group algebras, Comm. Algebra 38 (2010), no. 9, 3532--3538.
S. Akbari, A. Mohammadian, H. Radjavi and P. Raja, On the diameters of commuting graphs, Linear Algebra Appl. 418 (2006), no. 1, 161--176.
S. Akbari, H. Bidkhori and A. Mohammadian, Commuting graphs of matrix algebras, Comm. Algebra 36 (2008), no. 11, 4020--4031.
S. Akbari and P. Raja, Commuting graphs of some subsets in simple rings, Linear Algebra Appl. 416 (2006) 1038--1047.
J. Araujo, M. Kinyon and J. Konieczny, Minimal paths in the commuting graphs of semigroups, European J. Combin. 32 (2011), no. 2, 178--197.
G. Dolinar, B. Kuzma and P. Oblak, On maximal distances in a commuting graph, Electron. J. Linear Algebra 23 (2012) 243--256.
D. Dolzan, D. Kokol Bukovsek and P. Oblak, Diameters of commuting graphs of matrices over semirings, Semigroup Forum 84 (2012), no. 2, 365--373.
D. Dolzan and P. Oblak, Commuting graphs of matrices over semirings, Linear Algebra Appl. 435 (2011), no. 7, 1657--1665.
M. Giudici and A. Pope, The diameters of commuting graphs of linear groups and matrix rings over the integers modulo m, Australas. J. Combin. 48 (2010) 221--230.
C. Miguel, A note on a conjecture about commuting graphs, Linear Algebra Appl. 438 (2013), no. 12, 4750--4756.
A. Mohammadian, On commuting graphs of finite matrix rings, Comm. Algebra 38 (2010), no. 3, 988--994.
G. R. Omidi and E. Vatandoost, On the commuting graph of rings, J. Algebra Appl. 10 (2011), no. 3, 521--527.

### History

• Receive Date: 09 June 2014
• Revise Date: 03 November 2015
• Accept Date: 14 February 2017
• First Publish Date: 22 February 2017