scholarly journals The conjugacy class graphs of non-abelian 3-groups

2020 ◽  
Vol 16 (3) ◽  
pp. 297-299
Author(s):  
Athirah Zulkarnain ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Complete Graphs ◽  
Dominating Number ◽  
Minimum Number ◽  
Vertex Set ◽  
Class Graph

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element  is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to  if it satisfies  for some elements  in  with its inverse . A conjugacy class graph of a group   is defined when its vertex set is the set of non-central conjugacy classes of  . Two distinct vertices   and   are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of  and  have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a  -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

ON GRAPHS ASSOCIATED TO CONJUGACY CLASSES OF SOME THREE-GENERATOR GROUPS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Line Segments ◽  
Dominating Number ◽  
Commuting Graph ◽  
Independent Number ◽  
Graph Properties ◽  
Class Graph

A graph consists of points which are called vertices, and connections which are called edges, which are indicated by line segments or curves joining certain pairs of vertices.  In this paper, four types of graphs which are the commuting graph, non-commuting graph conjugate graph and the conjugacy class graph for some three-generator groups are discussed. Some of the graph properties are also found which include the independent number, chromatic number, clique number and dominating number.

THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ain Asyikin Ibrahim ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Dihedral Group ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Quaternion Group ◽  
Metabelian Groups ◽  
Dominating Number ◽  
Graph Properties ◽  
Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.   

scholarly journals General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

scholarly journals The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

2020 ◽  
Vol 7 (4) ◽  
pp. 62-71
Author(s):  
Zuzan Naaman Hassan ◽  
Nihad Titan Sarhan
Keyword(s):  
Conjugacy Class ◽  
Adjacency Matrix ◽  
Finite Graph ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Alternating Groups ◽  
Eigen Values ◽  
Finite Set ◽  
Class Graph ◽  
Square Matrix

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group  . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group  of some order for   and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

On the b-Chromatic Sum of Mycielskian of Km, n, Kn and Cn

2020 ◽  
Vol 20 (02) ◽  
pp. 2050007
Author(s):  
P. C. LISNA ◽  
M. S. SUNITHA
Keyword(s):  
Image Processing ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Color Class ◽  
Vertex Set ◽  
Chromatic Sum ◽  
Complete Bipartite Graphs

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

2-distance vertex-distinguishing total coloring of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850018
Author(s):  
Yafang Hu ◽  
Weifan Wang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Number Formula ◽  
Proper Total Coloring ◽  
Distance Formula

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

scholarly journals Recognition and Optimization Algorithms for P5-Free Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 304
Author(s):  
Mihai Talmaciu ◽  
Luminiţa Dumitriu ◽  
Ioan Şuşnea ◽  
Victor Lepin ◽  
László Barna Iantovics
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Independent Set ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Feedback Vertex Set ◽  
Free Graph ◽  
Dominating Number ◽  
Vertex Set ◽  
Np Complete

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

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