A PYTHON CODE FOR GENERATING ALL PROPER SUBGROUPS OF DIHEDRAL GROUP

The dihedral group of order 2n denoted by D_2n is the symmetry group of a regular -polygon consisting of rotation and reflection elements and the composition of both elements. Like any other group, the dihedral group also have a subgroup whose numbers differs depending on the value of n. This research is conducted by studying past literature and explore a new development to a theory. In this paper, all the form of proper subgroups of D_2n will be given and all of these proper subgroups of D_2n will be generated and counted with the help of Python program

The Intersection Graph of a Dihedral Group

The intersection graph of a finite group G is a graph (V,E) where V is a set of all non-trivial subgroups of G and E is a set of edges where two distinct subgroups H_i , H_j  are said to be adjacent if and only if H_i \cap H_j \neq {e} . This study discusses the intersection graph of a dihedral group D_{2n} specifically the subgraph, degree of vertices, radius, diameter, girth, and domination number. From this study, we obtained that if n=p^2 then the intersection graph of D_{2n} is containing complete subgraph K_{p+2} and \gamma(\Gamma_{D_{2n}})=p.

The Power Graph of a Dihedral Group

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

Topological Indices of the Relative Coprime Graph of the Dihedral Group

Assuming that G is a finite group and H is a subgroup of G, the graph known as the relative coprime graph of G with respect to H (denoted as Γ_(G,H)) has vertices corresponding to elements of G. Two distinct vertices x and y are adjacent by an edge if and only if (|x|,|y|)=1 and x or y belongs to H. This paper will focus on  finding the general formula for some topological indices of the relative coprime graph of a dihedral group. The study of topological indices in graph theory offers valuable insights into the structural properties of graphs. This study is conducted by reviewing many past literatures and then from there we infer a new result. The obtained outcomes will include measurements of distance, degree of vertex, and various topological indices such as the first Zagreb index, second Zagreb index, Wiener index, and Harary index that are associated with distance and degree of vertex

ON PROPERTIES OF PRIME IDEAL GRAPHS OF COMMUTATIVE RINGS

The prime ideal graph of  in a finite commutative ring  with unity, denoted by , is a graph with elements of  as its vertices and two elements in  are adjacent if their product is in . In this paper, we explore some interesting properties of . We determined some properties of  such as radius, diameter, degree of vertex, girth, clique number, chromatic number, independence number, and domination number. In addition to these properties, we study dimensions of prime ideal graphs, including metric dimension, local metric dimension, and partition dimension; furthermore, we examined topological indices such as atom bond connectivity index, Balaban index, Szeged index, and edge-Szeged index

SOME RESULT OF NON-COPRIME GRAPH OF INTEGERS MODULO n GROUP FOR n A PRIME POWER

One interesting topic in algebra and graph theory is a graph representation of a group, especially the representation of a group using a non-coprime graph.  In this paper, we describe the non-coprime graph of integers modulo  group and its subgroups, for  is a prime power or  is a product of two distinct primes

THE HARMONIC INDEX AND THE GUTMAN INDEX OF COPRIME GRAPH OF INTEGER GROUP MODULO WITH ORDER OF PRIME POWER

In the field of mathematics, there are many branches of study, especially in graph theory, mathematically a graph is a pair of sets, which consists of a non-empty set whose members are called vertices and a set of distinct unordered pairs called edges. One example of a graph from a group is a coprime graph, where a coprime graph is defined as a graph whose vertices are members of a group and two vertices with different x and y are neighbors if only if (|x|,|y|)=1. In this study, the author discusses the Harmonic Index and Gutman Index of Coprime Graph of Integer Group Modulo n. The method used in this research is a literature review and analysis based on patterns formed from several case studies for the value of n. The results obtained from this study are the coprime graph of the group of integers modulo n has the harmonic index of  and the Gutman index  for  where  is prime and  is a natural number

A Note on Nilpotent Graph of Ring Integer Modulo with Order Prime Power

Graf nilpoten dari gelanggang bilangan bulat modulo merupakan salah satu representasi graf pada struktur aljabar. Penelitian ini bertujuan mencari bentuk dan sifat graf nilpoten dari gelanggang bilangan prima modulo yang kemudian digeneralisasi menjadi gelanggang bilangan bulat modulo berpangkat prima sebarang. Metode yang digunakan pada penelitian ini adalah studi literatur. Pada gelanggang bilangan prima modulo, diperoleh bentuk graf nilpotennya adalah suatu graf bintang. Kemudian karakteristik dari graf nilpoten pada gelanggang bilangan bulat modulo berpangkat prima sebarang adalah memuat subgraf lengkap  dan memuat  buah subgraf bintang .Nilpotent graph of ring integer modulo is one of the graph representations in algebraic structures. This study aims to find out the shape and properties of a nilpotent graph of ring prime numbers modulo which is then generalized to a ring of integers modulo with arbitrary prime power. The method used in this research is a literature study. In the ring of integer modulo, we get the shape of a nilpotent graph as a star graph. Then, the characteristic of a nilpotent graph on a ring integer modulo with arbitrary prime power is that it contains a complete subgraph and contains a number of as a star subgraph