The topological indices of the non-commuting graph for symmetric groups

Topological indices are the numerical values that can be calculated from a graph and it is calculated based on the molecular graph of a chemical compound. It is often used in chemistry to analyse the physical properties of the molecule which can be represented as a graph with a set of vertices and edges. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if they do not commute. The symmetric group, denoted as, is a set of all permutation under composition. In this paper, two of the topological indices, namely the Wiener index and the Zagreb index of the non-commuting graph for symmetric groups of order 6 and 24 are determined

The Harary index of the non-commuting graph for dihedral groups

Assume G is a non-abelian group which consists a set of vertices, V = {v1 , v2, ..., vn} and a set of edges, E = {e1, e2, ..., em} where n and m are the positive integers. The non-commuting graph of G, denoted by ΓG, is the graph of vertex set G−Z(G), whose vertices are non-central elements, in which Z(G) is the center of G and two distinct vertices are adjacent if and only if they do not commute. In addition, the Harary index of a graph ΓG is the half-sum of the elements in the reciprocal distance of Dij where Dij the distance between vertex i and vertex j. In this paper, the Harary index of the non-commuting graph for dihedral groups is determined and its general formula is developed

The applications of zero divisors of some finite rings of matrices in probability and graph theory

Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph

The schur multiplier of pairs of nonabelian groups of order p4

Let (G,N) be a pair of groups where G is any group and N is a normal subgroup of G, then the Schur multiplier of pairs of groups is a functorial abelian group. The notion of the Schur multiplier of pairs of groups is an extension from the Schur multiplier of a group G. In this research, the Schur multiplier of pairs of finite nonabelian groups of order p4, where p is an odd prime, is determined

Energy of Cayley graphs for alternating 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 finite group G with respect to the subset S is a graph where the vertices are the elements of G and two vertices a and b in G are adjacent if ab−1 are in the set S. For a simple graph, the energy of a graph can be determined by its eigenvalues. Let Γ be a simple graph. Then by the summation of the absolute values of the eigenvalues of the adjacency matrix of the graph, its energy can be determined. This paper presents the Cayley graphs of alternating groups with respect to the subset S of valency 1 and 2. From the Cayley graphs, the eigenvalues are computed by using some properties of special graphs and then used to compute their energy

Regular ag-groupoids characterized by (∈, ∈ ∨ q k)-fuzzy ideals

In this paper, we introduce a considerable machinery which permits us to characterize a number of special (fuzzy) subsets in AG -groupoids. Generalizing the concepts of (∈, ∈ ∨q) -fuzzy bi-ideals (interior ideal), we define (∈, ∈ ∨ q k) -fuzzy bi-ideals, (∈, ∈ ∨ q k )-fuzzy left (right)-ideals and ( , ) k ? ? ?q -fuzzy interior ideals in AG -groupoids and discuss some fundamental aspects of these ideals in AG -groupoids. We further define ( ∈, ∈ ∨ q k) -fuzzy bi-ideals and (∈, ∈ ∨ q k)-fuzzy interior ideals and give some of their basic properties in AG -groupoids. In the last section, we define lower/upper parts of (∈, ∈ ∨ q k ) -fuzzy left (resp. right) ideals and investigate some characterizations of regular and intera-regular AG -groupoids in terms of the lower parts of ( ∈, ∈ ∨ q k ) -fuzzy left (resp. right) ideals and ( ∈, ∈ ∨ q k )-fuzzy bi-ideal of AG -groupoids

Independence polynomial of the commuting and noncommuting graphs associated to the quasidihedral group

An independence polynomial is a type of graph polynomial from graph theory that store combinatorial information such as the graph properties or graph invariants. The independence polynomial of a graph contains coefficients that represent the number of independent sets of certain sizes and the degree of the polynomial denotes the independence number of the graph. A graph of group G is called commuting graph if the vertices are noncentral elements of G and two vertices are adjacent if and only if they commute in G. Meanwhile, a noncommuting graph of a group G has a vertex set that contains all noncentral elements of G and two vertices are adjacent if and only if they do not commute in G. Since the group properties can be presented as graph from graph theory, then the graph polynomial of such graph should also be identified. Therefore, in this research, the independence polynomials are determined for the commuting and noncommuting graphs that are associated to the quasidihedral group

The squared Commutativity degree of dihedral groups

The commutativity degree of a finite group is the probability that a random pair of elements in the group commute. Furthermore, the n-th power commutativity degree of a group is a generalization of the commutativity degree of a group which is defined as the probability that the n-th power of a random pair of elements in the group commute. In this paper, the n-th power commutativity degree for some dihedral groups is computed for the case n equal to 2, called the squared commutativity degree

The central subgroup of the nonabelian tensor square of Bieberbach group with point group C2 C2

A Bieberbach group with point group  C2 xC2  is a free torsion crystallographic group. A central subgroup of a nonabelian tensor square of a group G, denoted by ∇(G) is a normal subgroup generated by generator g⊗g for all g∈G and essentially depends on the abelianization of the group. In this paper, the formula of the central subgroup of the nonabelian tensor square of one Bieberbach group with point group   C2 xC2 , of lowest dimension 3, denoted by S3 (3) is generalized up to n dimension. The consistent polycyclic presentation, the derived subgroup and the abelianization of group this group of n dimension are first determined. By using these presentations, the central subgroup of the nonabelian tensor square of this group of n dimension is constructed. The findings of this research can be further applied to compute the homological functors of this group.Keywords: Bieberbach group; central subgroup; nonabelian tensor square

Some considerations on the n-th commutativity degrees of finite groups

Let G be a finite group and n a positive integer. The n-th commutativity degree P-n(G) of G is the probability that the n-th power of a random element of G commutes with another random element of G. In 1968, P. Erdos and P.Turan investigated the case n = 1, involving only methods of combinatorics. Later several authors improved their studies and there is a growing literature on the topic in the last 10 years. We introduce the relative n-th commutativity degree P-n(H, G) of a subgroup H of G. This is the probability that an n-th power of a random element in H commutes with an element in G. The influence of P, (G) and P-n (H, G) on the structure of G is the purpose of the present work