The Power Graph of a Dihedral Group

Abstract

Graph theory is one of the topics in mathematics that is quite interesting to study because it is applicable and can be combined with other mathematical topics such as group theory. The combination of graph theory and group theory is that graphs can be used to represent a group. An example of a graph is a power graph. A power graph of the group is defined as a graph whose vertex set is all elements of and two distinct vertices and are connected if and only if or for a positive integer x and y. In this study, the author discusses the power graph of the dihedral group The results obtained from this study are the power graph of the dihedral group where with prime numbers and an natural number is a graph consisting of two non-disjoint subgraphs, namely complete subgraphs and star subgraphs. And we find that its radius and diameter are 1 and 2.

Numerical Invariants of Coprime Graph of A Generalized Quaternion Group

“…Definition 2.10. [11] Let G be a graph, complement metric dimension of G, denoted by dim(G) is the maximum cardinality of complement resolving a set of G.…”

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

“…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]. Some other graphs visualize are the power graph of groups [8] [9], and the intersection graph of groups [10].…”