The Intersection Graph of a Dihedral Group

Nurhabibah, Nurhabibah; Syarifudin, Abdul Gazir; Wardhana, I Gede Adhitya Wisnu; Aini, Qurrotul

doi:10.29303/emj.v4i2.119

Cited by 7 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…According to [10], we can construct a tree from a connected graph 𝐺 by eliminating some edges so that there is no cycle in graph 𝐺. In this paper, we use Maple 13 to determine the number of spanning trees of 𝑃𝐺(ℤ 𝑛 ).…”

Section: Resultsmentioning

confidence: 99%

“…Therefore, by deleting the edge that connects vertex 1 and 0, 𝑃𝐺(ℤ 𝑛 ) will be a disconnected graph, hence the vertex-connectivity of 𝑃𝐺(ℤ 𝑛 ) is 1. ∎ According to [10], a cycle of length 3 is called a triangle. The following theorem lets us determine the minimum number of triangles of 𝑃𝐺(ℤ 𝑛 ).…”

Section: Some Characteristics Of 𝑷𝑮(ℤ 𝒏 )mentioning

confidence: 99%

“…In graph theory, there is a term called a Hamiltonian graph. Graph 𝐺 is called a Hamiltonian if graph 𝐺 contains a Hamiltonian cycle [10]. According to [10], a cycle is called Hamiltonian if it passes every vertex of 𝐺 exactly once.…”

Section: Some Characteristics Of 𝑷𝑮(ℤ 𝒏 )mentioning

confidence: 99%

See 2 more Smart Citations

Some Characteristics of the Prime Graph of Integer Modulo Groups

Maulana

Wardhana

Switrayni

et al. 2023

InPrime:Ind.Jour.Pure.Applied.Math

View full text Add to dashboard Cite

The notion of the prime graph of a ring R was first introduced by Bhavanari, Kuncham, and Dasari in 2010. The prime graph of a ring R, denoted by PG(R) is a graph whose vertices are all elements of the ring, where two distinct vertices x and y are adjacent if and only if xRy = 0 or yRx = 0. In this paper, we study the forms and properties of the prime graph of integer modulo group, and some examples of the number of its spanning trees. In this paper, it is found that for all n, the maximum degree of vertices of PG(Z_n) is n-1 and the minimum degree of its vertices is 1. Then, we show that for all n, PG(Z_n) is neither a Hamiltonian graph nor an Eulerian graph. We also found some examples of the number of its spanning trees.Keywords: prime graph; spanning trees; Hamiltonian graph. AbstrakKonsep mengenai graf prima dari suatu gelanggang R pertama kali diperkenalkan oleh Bhavanari, Kuncham, dan Dasari pada tahun 2010. Graf prima dari suatu gelanggang R, yang dinotasikan dengan PG(R), adalah suatu graf yang simpul-simpulnya merupakan semua elemen dari gelanggang tersebut dengan dua buah simpul x dan y yang berbeda akan bertetangga jika dan hanya jika xRy = 0 atau yRx = 0. Di dalam penelitian ini, dikaji mengenai bentuk-bentuk dan sifat-sifat dari PG(Z_n), dan beberapa contoh dari banyak pohon pembangunnya. Pada penelitian ini, ditemukan hasil bahwa untuk setiap n, derajat maksimal dari simpul-simpul di PG(Z_n) adalah n-1 dan derajat minimum dari simpul-simpulnya adalah 1. Hasil selanjutnya yaitu, untuk setiap n, PG(Z_n) bukan merupakan suatu graf Hamiltonian atau graf Eulerian. Ditemukan juga beberapa contoh dari banyaknya pohon pembangun dari PG(Z_n).Kata Kunci: graf prima; pohon pembangun; graf Hamiltonian. 2020MSC: 05E16, 05C90, 20C05

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Some Characteristics Of 𝑷𝑮(ℤ 𝒏 )mentioning

confidence: 99%

See 1 more Smart Citation

Some Characteristics of the Prime Graph of Integer Modulo Groups

Maulana

Wardhana

Switrayni

et al. 2023

InPrime:Ind.Jour.Pure.Applied.Math

View full text Add to dashboard Cite

show abstract

“…In their paper, they examine the intersection graph of a group ℤ 𝑛 [11]. Later in 2021, Nurhabibah et al studied the intersection graph of the dihedral groups with prime square order, and give some properties on its shape, degree of vertices, radius, diameter, girth, and domination number [12]. In this article, we give a more general result of the intersection graph of the dihedral group with prime power order.…”

Section: Introductionmentioning

confidence: 96%

The Intersection Graph Representation of a Dihedral Group With Prime Order and Its Numerical Invariants

Ramdani

Wardhana

Awanis

2022

BAREKENG: J. Math. & App.

Self Cite

View full text Add to dashboard Cite

One of the concepts in mathematics that developing rapidly today is Graph Theory. The development of Graph Theory has been combined with Group Theory, that is by representing a group in a graph. The intersection graph from group , noted by , is a graph whose vertices are all non-trivial subgroups of group and two distinct vertices are adjacent in if and only if . In this research the intersection graph of a Dihedral group, we looking for the shapes and numerical invariants. The results obtained are if for , then has a subgraphs and subgraphs , the girth of the graph is 3, radius and diameter of the graph in a row is 2 and 3, and the chromatic number of the graph is

show abstract

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Husni

Syafitri

Siboro

et al. 2022

BAREKENG: J. Math. & App.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

The Intersection Graph of a Dihedral Group

Cited by 7 publications

References 0 publications

Some Characteristics of the Prime Graph of Integer Modulo Groups

Some Characteristics of the Prime Graph of Integer Modulo Groups

The Intersection Graph Representation of a Dihedral Group With Prime Order and Its Numerical Invariants

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

Contact Info

Product

Resources

About