The Third-Noncommuting Graph of a Group
DOI:
https://doi.org/10.5269/bspm.v34i1.23517Abstract
‎Let $ G $ be a group and let $T^{3}(G)$ be the proper subgroup $\lbrace h\in G \vert (gh)^{3}=(hg)^{3},~for~all‎~ ‎g\in G\rbrace $ of $ G $‎. ‎\textit{The third-noncommuting graph} of $ G $ is the graph with‎ vertex set $ G\setminus T^{3}(G) $‎, ‎where two vertices $ x $ and $ y $ are adjacent if $ (xy)^{3}\neq (yx)^{3} $‎. In this paper‎, ‎at first we obtain some results for this graph for any group $G$‎. ‎Then‎, ‎we investigate the structure of this graph for some groups‎.
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2. Abdollahi, A. and Taeri, B., Some conditions on infinite subsets of infinite groups, Bull. Malaysian Math. Soc. (Second Series) 22, 87-93, (1999).
3. Balakrishnan, R. and Ranganathan, K., A Text Book of Graph Theory, Springer, (1999).
4. Bondy, J. A. and Murty, J. S. R., Graph Theory with Applications, Elsevier, New York, (1977).
5. Darafsheh, M. R., Groups with the same non-commuting graph, Discrete Applied Mathematics, 157, 833-837, (2009).
6. Iranmanesh, A. and Jafarzadeh, A., Characterization of finite groups by their commuting graph, Acta Math. Academiae Paedagogicae Nyiregyhaziensis, 23, 7-13, (2007).
7. Mashkouri, M. and Taeri, B., On a graph associated to groups, Bull. Malays. Math. Sci. Soc., (2) 34(3), 553-560, (2011).
8. Neumann, B. H., A problem of Paul Erdos on groups, J. Aust. Math. Soc. Ser. A, 21. 467-472, (1976).
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Published
2015-05-06
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