General form of domination polynomial for two types of graphs associated to dihedral groups

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

The Wiener and Zagreb indices of conjugacy class graph of the dihedral groups

Topological indices are numerical values that can be analysed to predict the chemical properties of the molecular structure which are computed for a graph related to groups. Meanwhile, the conjugacy class graph of G is defined as a graph with a vertex set represented by the non-central conjugacy classes of G. Two distinct vertices are connected if they have a common prime divisor. The main objective of this article is to find various topological indices including the Wiener index, the first Zagreb index and the second Zagreb index for the conjugacy class graph of dihedral groups of order 2n where the dihedral group is the group of symmetries of regular polygon, which includes rotations and reflections. Many topological indices have been determined for simple and connected graphs in general but not graphs related to groups. In this article, the Wiener index and Zagreb index of conjugacy class graph of dihedral groups are generalized

The Laplacian energy of conjugacy class graph of some finite groups

Let G be a dihedral group and ΓdG its conjugacy class graph. The Laplacian energy of the graph, LE(ΓdG) is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined

The energy of four graphs of some metacyclic 2-groups

Let G be a metacyclic 2-group and gamma_G is the graph of G. The adjacency matrix of gamma_G is a matrix A=[a_ij] consisting of 0's and 1's in which the entry a_ij is 1 if there is an edge between the ith and jth vertices and 0 otherwise. The energy of a graph is the sum of all absolute values of the eigenvalues of the adjacency matrix of the graph. In this paper, the energy of commuting graph, non-commuting graph, conjugate graph and conjugacy class graph of metacyclic 2-groups are presented. The results show that the energy of these graphs of the groups must be an even integer

THE RELATIONSHIP BETWEEN ENTREPRENEURSHIP EDUCATION AND ENTREPRENEURIAL INTENTION OF UNIVERSITI KUALA LUMPUR – TEKNOPUTRA ALUMNI

Purpose: The unemployment rate among graduates is considered high in Malaysia despite concerted efforts taken by the Malaysian government in resolving the issue.  If this issue is not tackled immediately by the government, the unemployment rate may soar and hence lead to social problems in the country. The aim of this study is to explore the relationship between entrepreneurship education and entrepreneurial intention of Universiti Kuala Lumpur TEKNOPUTRA Alumni. Methodology: An online survey was conducted to explore if there is a relationship between entrepreneurship education the graduates had received at the university and their entrepreneurial intention to become entrepreneurs. About 50 graduates responded to the survey and they consisted of male and female graduates most of who were aged between 20 and 25 with less than 2 years of working experience. The sample had ventured into service types of business while others were sole proprietors, and most of them have been involved in their business for less than 5 years. Results: Cronbach Alpha values for the reliability analysis of items for entrepreneurship education (α=0.953) and entrepreneurial intention (α=0.893) show that the items are reliable. Inferential statistics, that is Pearson r correlation was run to determine the relationship between the independent variable, entrepreneurship education and the dependent variable entrepreneurship intention. Implications:  Pearson Correlation analysis shows that there is a positive and significant relationship between entrepreneurship educations with entrepreneurial intention among TEKNOPUTRA Alumni. In short, this implies that most of the TEKNOPUTRA Alumni agreed that entrepreneurship education they had received at the university has influenced their entrepreneurial intention to become entrepreneurs

The energy of cayley graphs for a generating subset of the dihedral groups

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a group G with respect to the subset S is a graph where its vertices are the elements of G and two vertices a and b are connected if ab^(−1) is in the subset S. The energy of a Cayley graph is the sum of all absolute values of the eigenvalues of its adjacency matrix. In this paper, we consider a specific subset S = {b, ab, . . . , a^(n−1)b} for dihedral group of order 2n, where n is greater or equal to 3 and find the Cayley graph with respect to the set. We also calculate the eigenvalues and compute the energy of the respected Cayley graphs. Finally, the generalization of the energy of the respected Cayley graphs is found

Generating finite cyclic and dihedral groups using sequential insertion systems with interactions

The operation of insertion has been studied extensively throughout the years for its impact in many areas of theoretical computer science such as DNA computing. First introduced as a generalization of the concatenation operation, many variants of insertion have been introduced, each with their own computational properties. In this paper, we introduce a new variant that enables the generation of some special types of groups called sequential insertion systems with interactions. We show that these new systems are able to generate all finite cyclic and dihedral groups

The Co-Prime Probability of p-Groups / Norarida Abdul Rhani ...[et al.]

The commutativity degree of Gcanbe used to measure how close a group is to be abelian. This concept has been extended to the probability that two random integers are co-prime. Two integers s and tare said to be co-prime if their greatest common divisor is equal to one. This concept hasbeen further extended to the co-prime probability of G, where the probability of the order of a random pair of elements in the group are co-prime.In this paper, the co-prime probability for all p-groups, where pis prime number isdetermine

Biomolecular aspects of second order limit language

The study on the recombinant behavior of double-stranded DNA molecules has led to the mathematical modelling of DNA splicing system. The interdisciplinary study is founded from the knowledge of informational macromolecules and formal language theory. A splicing language is resulted from a splicing system. Recently, second order limit language, a type of the splicing language, has been extensively explored. Before this, several types of splicing languages have been experimentally proven. Therefore, in this paper, a laboratory experiment was conducted to validate the existence of a second order limit language. To accomplish it, an initial strand of double-stranded DNA, amplified from bacteriophage lambda, was generated through polymerase chain reaction to generate thousands of copies of double-stranded DNA molecules. A restriction enzyme and ligase were added to the solution to complete the reaction. The reaction mixture was then subjected to polyacrylamide gel electrophoresis to separate biological macromolecules according to their sizes. A mathematical model derived at the early study was used to predict the approximate length of each string in the splicing language. The results obtained from the experiment are then used to verify the mathematical model of a second order limit language. This study shows that the theory on the second order limit language is biologically proven hence the model has been validated

The independence polynomial of n-th central graph of dihedral groups

An independent set of a graph is a set of pairwise non-adjacent vertices while the independence number is the maximum cardinality of an independent set in the graph. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called n-th central if the vertices are elements of G and two distinct vertices are adjacent if they are elements in the n-th term of the upper central series of G. In this research, the independence polynomial of the n-th central graph is found for some dihedral groups