## Main Article Content

## Abstract

The coprime graph of a finite group was defined by Ma, denoted by ΓG, is a graph with vertices that are all elements of group G and two distinct vertices x and y are adjacent if and only if (|x|, |y|) = 1. In this study, we discuss numerical invariants of a generalized quaternion group. The numerical invariant is a property of a graph in numerical value and that value is always the same on an isomorphic graph. This research is fundamental research and analysis based on patterns in some examples. Some results of this research are the independence number of ΓQ4n is 4n − 1 or 3n and its complement metric dimension is 4n − 2 for each n ≥ 2.

## Keywords

## Article Details

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*How to Cite*

*Journal of the Indonesian Mathematical Society*,

*29*(1), 36–44. https://doi.org/10.22342/jims.29.1.1245.36-44

* * References

- X. Ma, H. Wei, and L.Yang, “The Coprime Graph of a Group, “ International Journal of Group Theory,” 13-23(2014).
- A.G. Syarifudin, Nurhabibah, D.P. Malik, and I.G.A.W. Wardhana, “Some Characterizations of the Coprime Graph of Dihedral Group D2n,” Journal of Physics, 1-3 (2021).
- R. Juliana, Masriani, I.G.A.W. Wardahana, N.W. Switrayni, and Irwansyah, “Coprime Graph of Integer Modulo n Group and its Subgroup, “ Journal of Fundamental Mathematics and Applications, 15-18 (2020).
- Nurhabibah, A.G. Syarifudin, and I.G.A.W.Wardhana, “Some Results of The Coprime Graph of a Generalized Quaternion Group Q4n,” Indonesian Journal of Pure and Applied Mathematics, 29-33(2021).
- B. Ahmadi and H. Doostie, “On the Periods of 2-Steps General Fibonacci Sequences in a Generalized Quaternion Groups, ”Discrete Dynamics and Nature Society, 1-8 (2012).
- Zahidah, S., Mahanani, D. M., Oktaviana, K. L. (2021). Connectivity Indices Of Coprime Graph Ofgeneralized Quaternion Group. In Journal of The Indonesian Mathematical Society (Vol. 27, No. 3, 285–296).
- R. Diestel, Graph Theory. (Springer: 2000). [8] N. Murugesan and D.S. Nair, “The Domination and Independence of Some Cubic Bipartite Graphs,” Int. J. Contemp. Matth.Sciences, 611-618 (2011).
- G.Chartrand, L. Eroh, M.A.Johnson, and O.R. Oellerman, “Resolvability in Graphs and the Metric Dimension of a Graph,” Discrete Applied Mathematics, 99-113 (2000).
- L. Susilowati, A.R. Slamin, and Rosfiana, “The Complement Metric Dimension of Graphs and its Operations,”. Int. J. Civ. Eng. Technol. (IJCIET), 2386-2396 (2019).
- 11. Asmarani, E. Y., Syarifudin, A. G., Wardhana, I. G. A. W., and Switrayni, N. W. (2021). The Power Graph of a Dihedral Group. Eigen Mathematics Journal, 80-85.
- Misuki, W. U., Wardhana, I. G. A. W., Switrayni, N. W., and Irwansyah. (2021, February). Some results of non-coprime graph of the dihedral group D 2 n for na prime power. In AIP Conference Proceedings (Vol. 2329, No. 1, p. 020005).

#### References

X. Ma, H. Wei, and L.Yang, “The Coprime Graph of a Group, “ International Journal of Group Theory,” 13-23(2014).

A.G. Syarifudin, Nurhabibah, D.P. Malik, and I.G.A.W. Wardhana, “Some Characterizations of the Coprime Graph of Dihedral Group D2n,” Journal of Physics, 1-3 (2021).

R. Juliana, Masriani, I.G.A.W. Wardahana, N.W. Switrayni, and Irwansyah, “Coprime Graph of Integer Modulo n Group and its Subgroup, “ Journal of Fundamental Mathematics and Applications, 15-18 (2020).

Nurhabibah, A.G. Syarifudin, and I.G.A.W.Wardhana, “Some Results of The Coprime Graph of a Generalized Quaternion Group Q4n,” Indonesian Journal of Pure and Applied Mathematics, 29-33(2021).

B. Ahmadi and H. Doostie, “On the Periods of 2-Steps General Fibonacci Sequences in a Generalized Quaternion Groups, ”Discrete Dynamics and Nature Society, 1-8 (2012).

Zahidah, S., Mahanani, D. M., Oktaviana, K. L. (2021). Connectivity Indices Of Coprime Graph Ofgeneralized Quaternion Group. In Journal of The Indonesian Mathematical Society (Vol. 27, No. 3, 285–296).

R. Diestel, Graph Theory. (Springer: 2000). [8] N. Murugesan and D.S. Nair, “The Domination and Independence of Some Cubic Bipartite Graphs,” Int. J. Contemp. Matth.Sciences, 611-618 (2011).

G.Chartrand, L. Eroh, M.A.Johnson, and O.R. Oellerman, “Resolvability in Graphs and the Metric Dimension of a Graph,” Discrete Applied Mathematics, 99-113 (2000).

L. Susilowati, A.R. Slamin, and Rosfiana, “The Complement Metric Dimension of Graphs and its Operations,”. Int. J. Civ. Eng. Technol. (IJCIET), 2386-2396 (2019).

11. Asmarani, E. Y., Syarifudin, A. G., Wardhana, I. G. A. W., and Switrayni, N. W. (2021). The Power Graph of a Dihedral Group. Eigen Mathematics Journal, 80-85.

Misuki, W. U., Wardhana, I. G. A. W., Switrayni, N. W., and Irwansyah. (2021, February). Some results of non-coprime graph of the dihedral group D 2 n for na prime power. In AIP Conference Proceedings (Vol. 2329, No. 1, p. 020005).