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

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 AbstrakGraf 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

The Intersection Graph Representation of a Dihedral Group With Prime Order and Its Numerical Invariants

“…𝛽 0 = {𝑒, 𝑏} 𝛽 1 = {𝑒, 𝑎𝑏} 𝛽 2 = {𝑒, 𝑎 2 𝑏} 𝛽 3 = {𝑒, 𝑎 3 𝑏} 𝛽 4 = {𝑒, 𝑎 4 𝑏} 𝛽 5 = {𝑒, 𝑎 5 𝑏} 𝛽 6 = {𝑒, 𝑎 6 𝑏} 𝛽 7 = {𝑒, 𝑎 7 𝑏}…”

“…𝛾 10 = {𝑒, 𝑎 2 , 𝑎 4 , 𝑎 6 , 𝑏, 𝑎 2 𝑏, 𝑎 4 𝑏, 𝑎 6 𝑏} 𝛾 11 = {𝑒, 𝑎 2 , 𝑎 4 , 𝑎 6 , 𝑎𝑏, 𝑎 3 𝑏, 𝑎 5 𝑏, 𝑎 7 𝑏}…”

“…The visualization is given some properties like girth, diameter, chromatic number, clique number, and the shape of several groups like the dihedral group, the integer modulo group, or the generalized quaternion group. See [1][2] [3] [4] [5] [6] [7] [8] [9] [10] for more detail.…”

Numerical Invariants of Coprime Graph of A Generalized Quaternion Group

“…For a finite group G, we can represent G in some simple graphs such as the coprime graph, the non-coprime graph, the power graph, the intersection graph, the commuting graph, and others. See [1] [2] [4] [11] [12] for details. In 2014, Ma et al [1] introduced the coprime graph of a finite group G, where the vertex set of the graph is G and two distinct vertices x and y are adjacent if and only if the order of x and the order of y are relative primes [1].…”

The Harmonic Index and the Gutman Index of Coprime Graph of Integer Group Modulo With Order of Prime Power

“…In recent years, a graph is used to visualize the algebraic structure such as groups or rings. Some of the graphs visualizing groups are the coprime graph of dihedral groups [1] and quaternion groups [2] on the dihedral group [3]. This coprime graph is introduced by Ma [4], and later the dual of the coprime graph, called non-coprime, that introduced by Mansoori [5], which also studied integer modulo [6] and dihedral group [7].…”