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.

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).

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Coprime Graph is a geometric representation of a group in the form of undirectedgraph. The coprime graph of a group G, denoted by $\Gamma_G$ is a graph whose vertices are all elements of group G; and two distinct vertices a and b are adjacent if and only if $(|a|,|b|)=1$. In this paper, we study coprime graph of integers modulo n group and its subgroups. One of the results is if n is a prime number, then coprime graph of integers modulo n group is a bipartite graph.

The Study of algebraic structures, especially on graphs theory, leads to anew topics of research in recent years. In this paper, the algebraic structures that will be represented by a coprime graph are the dihedral group and its subgroups. The coprime graph of a group G, denoted by \Gamma_D_2n is a graph whose vertices are elements of G and two distinct vertices a and b are adjacent if only if (|a,|b|)=1. Some properties of the coprime graph of a dihedral group D_2n are obtained. One of the results is if n is prime then \Gamma_D_2n is a complete bipartite graph. Moreover, if n is the power of prime then \Gamma_D_2n is a multipartite graph.