The Clique Numbers and Chromatic Numbers of The Coprime Graph of a Dihedral Group

Abstract

The graph has many properties and characterizations. One interesting topic to discuss is the clique numbers and chromatic numbers. This research will determine the clique numbers and chromatic numbers of the coprime graph of the dihedral group. One of the main results is if n = 2k then the chromatics numbers of the coprime graph of the dihedral group are 2, and if n is odd composite numbers, then the clique numbers and chromatics numbers are (m + 2).

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

“…The degree of the graph is defined as follows. Definition 3 [13] The degree of a graph is the number of edges that are incident to the vertex. It is annotated as 𝑑𝑒𝑔(𝑎) for 𝑎 any vertice.…”

“…And the distance of the graph will be defined below. Definition 4 [13] The distance of a graph is the number of edges in a shortened path connecting them.…”

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.…”