Groups for which the noncommuting graph is a split graph

Document Type : Research Paper

Authors

1 K. N. Toosi University of Technology

2 K.N. Toosi University of Technology

Abstract

The noncommuting graph $\nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting
graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.

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References

[1] S. Foldes and P. L. Hammer, Split graphs, Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), Congressus Numerantium, Utilitas Math., Winnip eg, Man., 1977 311-315.
[2] I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, 92, American Mathematical So ciety, Provi-dence, RI, 2008.
[3] P. L. Hammer and B. Simeone, The splittance of a graph, Combinatorica, 1 (1981) 275-284.
[4] A. R. Moghaddamfar, W. J. Shi, W. Zhou and A. R. Zokayi, On noncommutative graphs asso ciated with a nite group, (Russian), translation in Siberian Math. J., 46 (2005) 325-332.
[5] B. H. Neumann, Groups whose elements have b ounded orders, J. London Math. Soc., 12 (1937) 195-198.
[6] D. B. West, Introduction to Graph Theory, Second Edition, Prentice Hall, Inc., Upp er Saddle River, NJ, 1996.

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History

  • Receive Date: 10 June 2015
  • Revise Date: 07 October 2015
  • Accept Date: 07 October 2015
  • Published Online: 01 March 2017

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