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Properties of tiny braids and the associated commuting graph

  • Azeem Haider
  • Usman AliEmail author
  • Moin A. Ansari
Article
  • 10 Downloads

Abstract

In this text, we focus on a subset (called the set of tiny braids) of factors of the Garside braid and generalize some known results related to tiny braids. These generalized results, along with some combinatorics, strengthen the existing relationship between this subset and Fibonacci numbers. We also associate a commuting graph with the subset and explore its fundamental identities, including its order, diameter, girth and degree-related properties.

Keywords

Fibonacci numbers Centralizer Commuting graph Positive braid Tiny braid 

Mathematics Subject Classification

11B39 05A15 05A05 

Notes

References

  1. 1.
    Ali, U., Azam, F., Javaid, I., Haider, A.: Braids with trivial simple centralizer. Algebra Colloq. 22(4), 561–566 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ali, U., Haider, A.: Centralizer of braids and Fibonacci numbers. Util. Math. 89, 289–296 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Artin, E.: Theory of braids. Ann. Math. 48, 101–126 (1947)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ashraf, R., Berceanu, B.: Simple braids, arXiv:1003.6014v1 [math.GT] (2010)
  5. 5.
    Ashraf, R., Berceanu, B., Riasat, A.: Fibonacci numbers and positive braids. Ars Combin. 122, 299–306 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bates, C., Bondy, D., Perkins, S., Rowley, P.: Commuting involution graphs for symmetric groups. J. Algebra 266(1), 133–153 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bundy, D.: The connectivity of commuting graphs. J. Combin. Theory Ser. A 113(6), 995–1007 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Garside, F.A.: The braid group and other groups. Quart. J. Math. Oxford Ser. (2) 20, 235–254 (1969)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Iranmanesh, A., Jafarzadeh, A.: On the commuting graph associated with the symmetric and alternating groups. J. Algebra Appl. 7(1), 129–146 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kassel, C., Turaev, V.: Braid Groups. Grad. Texts in Math. Springer, New York (2008)zbMATHGoogle Scholar
  11. 11.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (1996)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsJazan UniversityJazanSaudi Arabia
  2. 2.Centre for Advanced Studies in Pure and Applied MathematicsBahauddin Zakaria UniversityMultanPakistan
  3. 3.Institut de Mathématiques de Jussieu-Paris Rive GaucheParisFrance

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