Conditions on the Edges and Vertices of Non-commuting Graph

Authors

  • M. Jahandideh Department of Mathematics, College of Polymer, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran
  • M. R. Darafsheh School of Mathematics College of Science, University of Tehran, Tehran, Iran
  • N. H. Sarmin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • S. M. S. Omer Department of Mathematics, Faculty of Science, Benghazi University, Libya

DOI:

https://doi.org/10.11113/jt.v74.1964

Keywords:

Finite group, non-commuting graph

Abstract

Abstract - Let G􀡳 be a non- abelian finite group. The non-commuting graph ,􀪡is defined as a graph with a vertex set􀡳 − G-Z(G)􀢆in which two vertices x􀢞 and y􀢟 are joined if and only if xy􀢞􀢟 ≠ yx􀢟􀢞.  In this paper, we invest some results on the number of edges set , the degree of avertex of non-commuting graph and the number of conjugacy classes of a finite group. In order that if 􀪡􀡳non-commuting graph of H ≅ non - commuting graph of G􀪡􀡴,H 􀡴 is afinite group, then |G􀡳| = |H􀡴| .

References

Abdollahi, A., S. Akbari and H. R. Maimani. 2006. Non-Commuting Graph of a Group. J. Algebra. 298: 468–492.

Bondy, J. A and J. S. Murty. 1977. Graph Theory with Applications. American Elsevier Publishing Co, Inc.

Gustafson, W. H. 1973. What is the Probability That Two Group Elements Commute? The American Mathematical Monthly. 80(9): 1031–1034.

Neuman, B. H. 1976. Problem of Paul EdrÄos on Groups. J. Austral. Math.Soc. 21: 467–472.

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Published

2015-04-13

Issue

Section

Science and Engineering