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Some Results of The Coprime Graph of a Generalized Quaternion Group Q_4n

Nurhabibah Nurhabibah, Abdul Gazir Syarifudin, I Gede Adhitya Wisnu Wardhana

Abstract


Abstract

The Coprime graph is a graph from a finite group that is defined based on the order of each element of the group. In this research, we determine the coprime graph of generalized quaternion group Q_(4n) and its properties. The method used is to study literature and analyze by finding patterns based on some examples. The first result of this research is the form of the coprime graph of a generalized quaternion group Q_(4n) when n = 2^k, n an odd prime number, n an odd composite number, and n an even composite number. The next result is that the total of a cycle contained in the coprime graph of a generalized quaternion group Q_(4n) and cycle multiplicity when  is an odd prime number is p-1.

Keywords: Coprime graph, generalized quaternion group, order, path

 

Abstrak

Graf koprima merupakan graf dari dari suatu grup hingga yang didefiniskan berdasarkan orde dari masing-masing elemen grup tersebut. Pada penelitian ini akan dibahas tentang bentuk graf koprima dari grup generalized quaternion Q_(4n). Metode yang digunakan dalam penelitian ini adalah studi literatur dan melakukan analisis berdasarkan pola yang ditemukan dalam beberapa contoh. Adapun hasil pertama dari penelitian adalah bentuk graf koprima dari grup generalized quaternion Q_(4n) untuk kasus n = 2^k, n bilangan prima ganjil ganjil, n bilangan komposit ganjil dan n bilangan komposit genap. Hasil selanjutnya adalah total sikel pada graf koprima dari grup generalized quaternion dan multiplisitas sikel ketika  bilangan prima ganjil adalah p-1.

Kata kunci: Graf koprima, grup generalized quternion, orde


Keywords


coprime, generalized quarternion group

References


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A. G. Syarifuddin, Nurhabibah, D. P. Malik and I. G. A. W. Wardhana, “Some Characterizations of Coprime Graph of Dihedral group D2n,” Journal of Physics: Conference Series, vol. 1722, no. 1, pp. 1-4, 2021.

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Abdussakir, N. N. Azizah and Nofandika, Teori Graf, Malang: UIN Malang Press, 2009.

G Chartrand, D Geller and S Hedetniemi “Graph with Forbidden Subgraphs,” Journal of Combinatorial Theory Series B, vol. 10, no. 1, pp. 12-41, 1971.


Full Text: PDF

DOI: 10.15408/inprime.v3i1.19670

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