Radius, Diameter, Multiplisitas Sikel, dan Dimensi Metrik Graf Komuting dari Grup Dihedral

Abstract

AbstrakGraf komuting C(G) dari suatu grup non Abelian G adalah graf yang memuat semua anggota grup G sebagai himpunan titik dan dua titik akan terhubung langsung di C(G) jika keduanya komutatif di grup G. Dalam artikel ini dibahas mengenai graf komuting dari grup dihedral D2n. Pembahasan dibatasi pada topik radius, diameter, multiplisitas sikel, dan dimensi metrik. Beberapa teorema disajikan terkait topik tersebut disertai bukti kebenarannya. Kata kunci: radius, diameter, multiplisitas sikel, dimensi metrik, graf komuting, grup dihedral. AbstractCommuting graph C(G) of a non Abelian group G is graph that contains all element of G as its vertex set and two distinct vertices in C(G) will be adjacent if they are commute in G. In this paper we discuss about commuting graph of dihedral group D2n. We show radius, diameter, cycle multiplicity, and metric dimension of this commuting graph in several theorems with their proof.

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}…”

“…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.…”

“…Based on Figure1, it is known that the shape of the graph 𝛤 𝐷 8 is in the form of a subgraph 𝐾 4 and 4 subgraph 𝐾 2 .Graph 𝛤 𝐷 2𝑛 for 𝑛 = 23 …”

Pelabelan Graf dalam Rencana Tata Ruang Rumah Sakit Islam Klaten

“…Sekarang semakin banyak muncul penggunaan model matematika maupun penalaran matematika sebagai alat bantu dalam menyelesaikan permasalahan yang dihadapi dalam berbagai disiplin ilmu. Dengan mengkaji dan menganalisa model atau rumusan teori graf dapat diperlihatkan peranan dan kegunaannya dalam memecahkan permasalahan [9][10] [11]. Salah satunya adalah masalah rencana tata ruang rumah sakit.…”

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

“…In 2010, Vahidi and Talebi [2] researched the properties of the commuting graph of the dihedral group and quaternion group and got some parameters of graph theory. In 2017, Abdussakir [3] researched the commuting graph of a dihedral group that is matrix dimension, cycle multiplicity, radius, and diameter. In the next year, Syarifuddin et al [4].…”