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

ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP

The coprime graph of a finite group , denoted by , is a graph with vertex set  such that two distinct vertices  and  are adjacent if and only if their orders are coprime, i.e.,  where |x| is the order of x. In this paper, we complete the form of the coprime graph of a dihedral group that was given by previous research and it has been proved that  if , for some  and  if . Moreover, we prove that if  is even, then the independence number of  is , where  is the greatest odd divisor of  and if  is odd, then the independence number of  is . Furthermore, the Wiener index of coprime graph of dihedral group has been stated here

On Prime Index of a Graph

In prime labeling, vertices are labeled from 1 to n, with the condition that any two adjacent vertices have relatively prime labels. Coprime labeling maintains the same criterion as prime labeling with adjacent vertices using any set of distinct positive integers. Minimum coprime number prG, is the minimum value k for which G has coprime labeling. There are many graphs that do not possess prime labeling, and hence have coprime labeling. The primary purpose of this work is to change a coprime labeled graph into a prime graph by removing the minimum number of edges. Thus, the prime index ε(G) is the least number of edges to be removed from a coprime graph G to form a prime graph. In this study, the prime index of various graphs is determined. Also, an algorithmic way to determine the prime index of the complete graph is found

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

Indeks eksentrisitas Zagreb pertama dan kedua graf koprima dari grup matriks upper unitriangular atas ring bilangan bulat modulo prima

INDONESIA: Graf koprima dari suatu grup G adalah graf Γ_G dengan G sebagai himpunan titiknya dan dua titik berbeda terhubung langsung jika dan hanya jika orde keduanya relatif prima. Misalkan p adalah bilangan prima, maka G_p melambangkan grup perkalian matriks 2×2 upper unitriangular atas ring bilangan bulat modulo p. Penelitian ini bertujuan untuk mengetahui graf koprima Γ_(G_p ) serta menentukan indeks eksentrisitas Zagreb pertama dan kedua dari graf koprima Γ_(G_p ) untuk p≥3. Metode yang digunakan dalam penelitian ini adalah penelitian kepustakaan (library research). Hasil dari penelitian ini adalah sebagai berikut. 1. Indeks eksentrisitas Zagreb pertama dari graf koprima Γ_(G_p )adalah E_1 (Γ_(G_p ))=4p-3. 2. Indeks eksentrisitas Zagreb kedua dari graf koprima Γ_(G_p ) adalah E_2 (Γ_(G_p ))=2p-2. ENGLISH: Coprime graph of a group G is graph Γ_G with G is as its set of vertices and the two distinct vertices are adjacent if and only if order of both of them are relatively prime. Let p is a prime number, then G_p denote the multiplicative group of 2×2 upper unitriangular matrices over ring of integers modulo p. The purposes of this research are to find out coprime graph Γ_(G_p ) and then find first and second Zagreb eccentricity indices of coprime graph Γ_(G_p ) for p≥3. The research method that used is library research. The results of this research are as follows. 1. First Zagreb eccentricity index of coprime graph Γ_(G_p )is E_1 (Γ_(G_p ))=4p-3. 2. Second Zagreb eccentricity index of coprime graph Γ_(G_p )is E_2 (Γ_(G_p ))=2p-2. ARABIC: البيان coprime لزمرة G هو البيان Γ_G حيث كانت G مجموعة الرؤوس من Γ_G والرؤوسان المختلفتان مجاوران إذا وفقط إذا كانت رتبة كل منهما أوليا نسبيا. دع p يكون عدد أولي، إذا G_p دلت زمرة الضربي من المصفوفات 2×2 upper unitriangular على حلقة الأعداد الصحيحة modulo prime. الأغراض في هذا البحث هي لمعرفة البيان coprime Γ_(G_p ) ووجد مؤشر الإنحراف الزغرب الأول والثاني على البيان coprime Γ_(G_p ) مع p≥3. طريقة البحث المستخدمة هنا هي دراسة مكتبيّة. وأما نتائج هذا البحث هي: ۱. مؤشر الإنحراف الزغرب الأول من البيان coprime Γ_(G_p ) هو E_1 (Γ_(G_p ))=4p-3. ۲. مؤشر الإنحراف الزغرب الثاني من البيان coprime Γ_(G_p ) هو E_2 (Γ_(G_p ))=2p-2

Quotients and subgroups of Baumslag-Solitar groups

We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).Comment: Final version, to appear in Journal of Group Theor

Local-global principles for norm one tori over semi-global fields

Let K be a complete discretely valued field with residue field k and F be a function field of a curve over K. Let L/F be a Galois extension of degree n. If n is coprime to char(k), then under some assumptions on k(e.g. k is algebraically closed or a finite field) and on the geometry of the curve, we show that there is a local-global principle with respect to discrete valuations for the norms from the extension L/F

Tropical secant graphs of monomial curves

The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular, old Sections 4 and 5 are merged into a single section. Also, added Figure 3 and discussed Chow polytopes of rational normal curves in Section