Vol 20 No 1 (2026): BAREKENG: Journal of Mathematics and Its Application
Articles

GRAPH ENERGY OF THE COPRIME GRAPH ON GENERALIZED QUATERNION GROUP

PDF
  • Miftahurrahman Miftahurrahman,
  • I Gede Adhitya Wisnu Wardhana,
  • Nur Idayu Alimon,
  • Nor Haniza Sarmin
Miftahurrahman Miftahurrahman
Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Mataram, Indonesia
I Gede Adhitya Wisnu Wardhana
Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Mataram, Indonesia
Nur Idayu Alimon
College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor, Malaysia
Nor Haniza Sarmin
Departement of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Malaysia
DOI: https://doi.org/10.30598/barekengvol20iss1pp0031-0040
Published November 24, 2025
Keywords
  • Coprime graph,
  • Graph energy,
  • Graph theory,
  • Quaternion group
How to Cite
[1]
M. Miftahurrahman, I. G. A. Wisnu Wardhana, N. I. Alimon, and N. H. Sarmin, “GRAPH ENERGY OF THE COPRIME GRAPH ON GENERALIZED QUATERNION GROUP”, BAREKENG: J. Math. & App., vol. 20, no. 1, pp. 0031-0040, Nov. 2025.

Copyright (c) 2025 Miftahurrahman Miftahurrahman, I Gede Adhitya Wisnu Wardhana, Nur Idayu Alimon, Nor Haniza Sarmin

Creative Commons License

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

Abstract

This paper investigates the Degree Square Sum Energy , Degree Exponent Energy , and Degree Exponent Sum Energy  of the coprime graph on generalized quaternion group ​. This research is quantitative study using previous study as the literature review to construct the new theorem. These energy methods provide new insights into the spectral properties of graphs by their vertex degree distributions into eigenvalue computations. Using spectral graph theory, the general formulas for the , , and  of  ​ are formulated for  for every positive integer . Furthermore, we explore the implications of these methods in understanding the algebraic and spectral characteristics of  ​. Numerical results are presented for specific cases to validate the previous theorem. This study contributes to the broader analysis of graph energies, offering a framework for studying other algebraic structures.

PDF

Downloads

Download data is not yet available.

References

  1. 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.
  2. N. Nurhabibah, I. G. A. W. Wardhana, and N. W. Switrayni, “NUMERICAL INVARIANTS OF COPRIME GRAPH OF A GENERALIZED QUATERNION GROUP,” Journal of the Indonesian Mathematical Society, vol. 29, no. 01, pp. 36–44, 2023.doi: https://doi.org/10.22342/jims.29.1.1245.36-44
  3. H. B. Shelash and M. Jasim, “CO-PRIME GRAPH OF FINITE GROUPS”, doi: 10.13140/RG.2.2.25739.41762.
  4. Nurhabibah, D. P. Malik, H. Syafitri, and I. G. A. W. Wardhana, “SOME RESULTS OF THE NON-COPRIME GRAPH OF A GENERALIZED QUATERNION GROUP FOR SOME N,” AIP Conf Proc, vol. 2641, no. December 2022, p. 020001, 2022, doi: https://doi.org/10.1063/5.0114975.
  5. M. Afdhaluzzikri, I. Gede, A. Wisnu Wardhana, F. Maulana, and H. R. Biswas, “THE NON-COPRIME GRAPHS OF UPPER UNITRIANGULAR MATRIX GROUPS OVER THE RING OF INTEGER MODULO WITH PRIME ORDER AND THEIR TOPOLOGICAL INDICES,” BAREKENG: J. Math. & App, vol. 19, no. 1, pp. 547–0556, 2025, doi: https://doi.org/10.30598/barekengvol19iss1pp547-556.
  6. A. R. Nagalakshmi, A. S. Shrikanth, G. K. Kalavathi, and K. S. Sreekeshava, “THE DEGREE ENERGY OF A GRAPH,” Mathematics, vol. 12, no. 17, Sep. 2024, doi: https://doi.org/10.3390/math12172699.
  7. B. B. and C. E., “DEGREE SQUARE SUM EQUIENERGETIC AND HYPERENERGETIC GRAPHS,” Malaya Journal of Matematik, vol. 8, no. 2, pp. 301–305, Apr. 2020, doi: https://doi.org/10.26637/MJM0802/0001.
  8. B. B. and C. E., “DEGREE SQUARE SUM EQUIENERGETIC AND HYPERENERGETIC GRAPHS,” Malaya Journal of Matematik, vol. 8, no. 2, pp. 301–305, Apr. 2020, doi: https://doi.org/10.26637/MJM0802/0001.
  9. H. S. Ramane and S. S. Shinde, “DEGREE EXPONENT POLYNOMIAL OF GRAPHS OBTAINED BY SOME GRAPH OPERATIONS,” Electron Notes Discrete Math, vol. 63, pp. 161–168, 2017.doi: https://doi.org/10.1016/j.endm.2017.11.010
  10. B. Basavanagoud and C. Eshwarachandra, “DEGREE EXPONENT SUM ENERGY OF A GRAPH,” 2020.doi: https://doi.org/10.56947/gjom.v8i1.306
  11. I. Gutman, X. Li, and J. Zhang, “GRAPH ENERGY,” Analysis of Complex Networks: From Biology to Linguistics, pp. 145–174, 2009.doi: https://doi.org/10.1002/9783527627981.ch7
  12. B. Basavanagoud and E. Chitra, “DEGREE SQUARE SUM ENERGY OF GRAPHS,” International Journal of Mathematics And its Applications, vol. 6, no. B, pp. 193–205, 2018, [Online]. Available: http://ijmaa.in/
  13. B. B. and C. E., “DEGREE SQUARE SUM EQUIENERGETIC AND HYPERENERGETIC GRAPHS,” Malaya Journal of Matematik, vol. 8, no. 2, pp. 301–305, Apr. 2020, doi: https://doi.org/10.26637/MJM0802/0001.
  14. S. R. Jog and J. R. Gurjar, “BOUNDS ON DEGREE SQUARE SUM DISTANCE SQUARE ENERGY OF GRAPHS,” BULLETIN Bull. Int. Math. Virtual Inst, vol. 11, no. 2, pp. 367–373, 2021, doi: 10.7251/BIMVI2102367J.
  15. P. Mahalank, H. S. Ramane, and A. R. Desai, “DEGREE EXPONENT ADJACENCY POLYNOMIAL OF SOME GRAPHS,” Adv. Math., Sci. J, vol. 9, no. 3, pp. 1001–1008, 2020.doi: https://doi.org/10.37418/amsj.9.3.25
  16. I. Gutman, “DEGREE-BASED TOPOLOGICAL INDICES,” Croatica Chemica Acta, vol. 86, no. 4, pp. 351–361, Dec. 2013, doi: https://doi.org/10.5562/cca2294.
  17. M. U. Romdhini, A. Nawawi, and C. Chuei Yee, “DEGREE EXPONENT SUM ENERGY OF COMMUTING GRAPH FOR DIHEDRAL GROUPS,” Malaysian Journal of Science, vol. 41, pp. 40–46, Sep. 2022, doi: https://doi.org/10.22452/mjs.sp2022no1.6.
  18. B. Ahmadi and H. Doostie, “ON THE PERIODS OF 2-STEP GENERAL FIBONACCI SEQUENCES IN THE GENERALIZED QUATERNION GROUPS,” Discrete Dyn Nat Soc, vol. 2012, 2012, doi: https://doi.org/10.1155/2012/458964.
  19. X. Ma, H. Wei, and L. Yang, “THE COPRIME GRAPH OF A GROUP Communicated by Mehri Akhavan-Malayeri,” 2014. [Online]. Available: www.ui.ac.ir
  20. H. R. Dorbidi, “A NOTE ON THE COPRIME GRAPH OF A GROUP Communicated by Alireza Abdollahi,” 2016. [Online]. Available: www.ui.ac.ir
  21. R. Balakrishnan, “THE ENERGY OF A GRAPH,” Linear Algebra Appl, vol. 387, no. 1-3 SUPPL., pp. 287–295, Aug. 2004, doi: https://doi.org/10.1016/j.laa.2004.02.038.
  22. H. S. Ramane, S. Shinde, and S. S. Shinde, “DEGREE EXPONENT POLYNOMIAL AND DEGREE EXPONENT ENERGY OF GRAPHS.” [Online]. Available: https://www.researchgate.net/publication/338195221
  23. B. Basavanagoud and C. Eshwarachandra, “DEGREE EXPONENT SUM ENERGY OF A GRAPH,” 2020.doi: https://doi.org/10.56947/gjom.v8i1.306
  24. N. Nurhabibah, A. G. Syarifudin, and I. G. A. W. Wardhana, “SOME RESULTS OF THE COPRIME GRAPH OF A GENERALIZED QUATERNION GROUP Q_4N,” InPrime: Indonesian Journal of Pure and Applied Mathematics, vol. 3, no. 1, pp. 29–33, 2021.doi: https://doi.org/10.15408/inprime.v3i1.19670
  25. H. S. Ramane and S. S. Shinde, “DEGREE EXPONENT POLYNOMIAL OF GRAPHS OBTAINED BY SOME GRAPH OPERATIONS,” Electron Notes Discrete Math, vol. 63, pp. 161–168, 2017. Doi: https://doi.org/10.1016/j.endm.2017.11.010