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Vol 17 No 3 (2023): BAREKENG: Journal of Mathematics and Its Applications
Articles

ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP

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  • Agista Surya Bawana,
  • Aluysius Sutjijana,
  • Yeni Susanti
Agista Surya Bawana
Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Gadjah Mada, Indonesia
Aluysius Sutjijana
Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Gadjah Mada, Indonesia
Yeni Susanti
Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Gadjah Mada, Indonesia
DOI: https://doi.org/10.30598/barekengvol17iss3pp1695-1702
Published September 30, 2023
Keywords
  • dihedral groups,
  • coprime graph,
  • girth,
  • independence number,
  • Wiener index
How to Cite
[1]
A. Bawana, A. Sutjijana, and Y. Susanti, “ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP”, BAREKENG: J. Math. & App., vol. 17, no. 3, pp. 1695-1702, Sep. 2023.

Copyright (c) 2023 Agista Surya Bawana, Aluysius Sutjijana, Yeni Susanti

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This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Abstract

The coprime graph of a finite group , denoted by , is a graph with vertex set  such that two distinct vertices  and  are adjacent if and only if their orders are coprime, i.e.,  where |x| is the order of x. In this paper, we complete the form of the coprime graph of a dihedral group that was given by previous research and it has been proved that  if , for some  and  if . Moreover, we prove that if  is even, then the independence number of  is , where  is the greatest odd divisor of  and if  is odd, then the independence number of  is . Furthermore, the Wiener index of coprime graph of dihedral group has been stated here.

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References

  1. R. J. Wilson, Introduction to Graph Theory, Fifth Edition, England: Pearson Education Limited, 2010.
  2. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Second Edition, New York: Springer Science+Business Media, 2012.
  3. W. D. Wallis, A Beginner's Guide to Graph Theory, Second Edition, Boston: Birkhauser, 2000.
  4. D. S. Dummit, and R. M. Foote, Abstract Algebra, Second Edition, New York: John Wiley and Sons, Inc., 1999.
  5. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, International Edition, Singapore: McGraw-Hill, 1997.
  6. J. Vahidi, and A. Asghar Talebi, “The Commuting Graphs on Groups D_2n and Q_n,” Journal of Mathematics and Computer Science, vol. 1, no. 2, pp. 123-127, 2010, doi: 10.22436/jmcs.001.02.07.
  7. X. L. Ma, H. Q. Wei, and G. Zhong, "The Cyclic Graph of a Finite Group," Algebra, vol. 2013, Article ID 107265, 7 pages, 2013, doi: 10.1155/2013/107265.
  8. M. Ghorbani, M. R. Darafsheh, and Pedram Yousefzadeh, “On the Prime Graph of a Finite Group,” Miskolc Mathematical Notes, vol. 22, no. 1, pp. 201–210, 2021, doi: 10.18514/MMN.2021.1668.
  9. F. Mansoori, A. Erfanian, and B. Tolue, “Non-coprime Graph of a Finite Group.” in AIP Conference Proceedings, 1750 (1): 050017, Jun. 21, 2016, doi: 10.1063/1.4954605.
  10. X. Ma, H. Wei, and L. Yang, “The Coprime Graph of a Group,” International Journal of Group Theory, vol. 3, no. 3, pp. 13-23, 2014, doi: 10.22108/IJGT.2014.4363.
  11. H. R. Dorbidi, “A Note on The Coprime Graph of a Group,” International Journal of Group Theory, vol. 5, no. 4, pp. 17-22, 2016, doi: 10.22108/IJGT.2016.9125.
  12. N. I. Alimon, N. H. Sarmin, and A. Erfanian, “The Szeged and Wiener Indices for Coprime Graph of Dihedral Groups,” in AIP Conference Proceedings 2266, 060006, Oct. 6, 2020, doi: 10.1063/5.0018270.
  13. S. A. Gazir, I. G. A. W. Wardhana, N. W. Switrayni, and Q. Aini, “Some Properties of Coprime Graph of Dihedral Group D_2n When n is a Prime Power,” Journal of Fundamental Mathematics and Applications, vol. 3, no. 1, pp. 34-38, 2020, doi: 10.14710/jfma.v3i1.7413.
  14. A. G. Syarifudin, Nurhabibah, D. P. Malik, and I. G. A. W. Wardhana, “Some Characterization of Coprime Graph of Dihedral Group D_2n,” in Journal of Physics: Conference Series, 1722 012051, 2021, doi: 10.1088/1742-6596/1722/1/012051.
  15. J. Hamm, and A. Way, “Parameters of the Coprime Graph of a Group,” International Journal of Group Theory, vol. 10, no. 3, pp. 137-147, 2021, doi: 10.22108/IJGT.2020.112121.1489.