Some graphs of metabelian groups of order 24 and their energy

The energy of a graph G is the sum of all absolute values of the eigenvalues of the adjacency matrix. An adjacency matrix is a square matrix where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. A commuting graph of a group is a graph whose vertex set is the non-central elements of the group and whose edges are pairs of vertices that commute. Meanwhile, a noncommuting graph is a graph whose vertex set is the non-central elements of the group but the edges are the pairs of vertices that do not commute. A conjugacy class graph is a graph with the non-central conjugacy classes vertices. Two vertices are connected if the order of the conjugacy classes have a common prime divisor. Besides, a conjugate graph is a graph whose vertex set is the non-central elements of the group where two distinct vertices are joined if they are conjugate. Furthermore, a group G is said to be metabelian if there exists a normal subgroup H in G such that both H and the factor group G/H are abelian. In this research, the energies of commuting graphs, noncommuting graphs, conjugacy class graphs and conjugate graphs for all nonabelian metabelian group of order 24 are determined. The computations of the graphs and adjacency matrices for the energy of graphs are determined with the assistance of Groups, Algorithms and Programming (GAP) and Maple 2016 softwares. The results show that the energy of graphs of the groups in the study must be an even integer in the case that the energy is rational

Mathematical analysis for tumor growth model of ordinary differential equations

Special functions occur quite frequently in mathematical analysis and lend itself rather frequently in physical and engineering applications. Among the special functions, gamma function seemed to be widely used. The purpose of this thesis is to analyse the various properties of gamma function and use these properties and its definition to derive and tackle some integration problem which occur quite frequently in applications. It should be noted that if elementary techniques such as substitution and integration by parts were used to tackle most of the integration problems, then we will end up with frustration. Due to this, importance of gamma function cannot be denied

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

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

Maple computations on the energy of cayley graphs for dihedral groups

Maple software is very useful in assisting various types of mathematical computations. The software provides many packages that can be explored such as Group Theory package, Graph Theory package and Linear Algebra package. The group theory package provides a collection of commands for computing and visualizing finitely generated groups. Meanwhile, the Graph Theory package is a collection of commands for creating graphs, drawing graphs, manipulating graphs and testing graphs for certain properties. The graphs are consists of sets of vertices which are connected by edges. The package supports both directed and undirected graphs but in this research, only undirected graphs are being considered. The aim of this paper is to present the Maple computation of the energy of Cayley graphs for a subset of the dihedral groups. The computation includes the construction of the Cayley graphs followed by the formation of their adjacency matrix and the computation of their eigenvalues. Lastly, the energy of the Cayley graphs can be found by calculating the summation of the absolute values of the eigenvalues found