The Structures of Non-Coprime Graphs for Finite Groups from Dihedral Groups with Regular Composite Orders
Abstract
For any finite group, the non-coprime graph of the group is a graph with vertices consisting of all non-identity elements of the group. Two different vertices are considered adjacent if their orders are not coprime, meaning their greatest common divisor (gcd) is not equal to one. Misuki provides the structure of the non-coprime graph for the dihedral group when the order is a prime power. We establish a more general property for cases where the order of the group is a regular composite number, discovering that the structure of the non-coprime graph for a dihedral group can be partitioned into some number of complete subgraphs.
Keywords: dihedral group; disjoint subgraph; non-coprime graph.
Abstrak
Untuk sebarang grup hingga, graf non-coprime dari grup tersebut adalah graf dengan simpul yang terdiri dari semua elemen non-identitas dari grup tersebut. Dua simpul yang berbeda dianggap bertetangga jika orde mereka tidak saling prima, artinya pembagi terbesar bersama (gcd) mereka tidak sama dengan satu. Misuki memberikan struktur graf non-coprime untuk grup dihedral ketika ordenya adalah pangkat prima. Pada studi ini didapatkan sifat yang lebih umum ketika orde grup adalah bilangan komposit biasa. Didapatkan juga bahwa struktur graf non-coprime dari grup dihedral dapat dipartisi menjadi beberapa subgraf lengkap.
Kata Kunci: graf non-koprima; grup dihedral; subgraph disjoin.
2020MSC: 05C25, 05C69, 20D60.
Keywords
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DOI: 10.15408/inprime.v5i2.29018
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