Verification of an old conjecture on nonabelian 2–generated groups of order p3

A longstanding conjecture in group theory states: "Every finite non-abelian p-group possesses at least a non-inner automorphism of order p", where p is a prime number. Recently, an updated classification of 2-generated p-groups of nilpotency class two has been published. Using this classification, we prove the verification of this conjecture for 2-generated groups of order p3

The conjugate graph and conjugacy class graph of order at most 32

A groupis called metacyclic if it has a cyclic normal subgroup such that the quotient groupis also cyclic. The classification of non-Abelianmetacyclicp-groups of class two has been found by earlier researcher, which is partitioned into two families of non-isomorphic p-groups. The conjugacy classes of these groups are then applied into graph theory. The conjugate graph is a graph whose the vertices are non-central elements of a finite non-Abelian group. Besides, the conjugacy class graph is a graph whose vertices are non-central of a group that is two vertices are connected if their cardinalities are not coprime, in which their greatest common divisor between the vertices is not equal to one. In this study, the conjugacy classes of the metacyclic 2-groups of order at most 32 have been obtained using the definition of conjugacy classes and their group presentations. The conjugate graph and conjugacy class graph of metacyclic 2-groups of order at most 32 are found directly using the definition. These conjugate graph and conjugacy class graph are then used to determine some graph properties such as chromatic number, clique number, dominating number and independent number. The conjugate graph of the groups turned out to be union of complete components of K2, meanwhile the conjugacy class graph of the groups turned out to be a complete graph

Probabilistic characterizations of some finite ring of matrices and its zero divisor graph

Let R be a finite ring. In this study, the probability that two random elements chosen from a finite ring have product zero is determined for some finite ring of matrices over Zn. Then, the results are used to construct the zero divisor graph which is defined as a graph whose vertices are the nonzero zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0

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

Partitioning Technique for Transformation of Connected Graphs into Single-Row Network

In this paper, we present our work called Connected Graph Sequence (CGS) which transforms a partially dense graph into the single-row network. Partially dense graph is a graph where a number of connected components, namely subgraphs, are connected by some links and each subgraph has a higher density value compare to the graph. The transformation is necessary in applications such as in the assignment of telephone channels to caller-receiver pairs roaming in cells in a cellular network on real-time basis. In this channel assignment application, each caller and receiver in a call is treated as a node, while their pair connection is treated as the edge. A specific case of the graph in the form of a partially dense graph is then transformed into its corresponding single-row network for assigning the channels to the caller-receiver pairs

The non-normal subgroup graph of some dihedral groups

Let be a finite groupand is a subgroup of. The subgroup graph of in is defined as a directed simple graph with vertex set and two distinct elements and are adjacent if and only if ∈. In this paper, the work on subgroup graph is extended by defining a new graph called the non-normal subgroup graph. The subgroup graph is determined for some dihedral groupsof order 2 when the subgroup is non-normal

Ordered semigroups characterized by (ϵ ϵ vqk)-fuzzy generalized bi-ideals

In this paper, we introduce a considerable machinery that permits us to characterize a number of special (fuzzy) subsets in ordered semigroups. In this regard, we generalize (Davvaz and Khan in Inform Sci 181:1759-1770 2011) and define (is an element of, is an element of boolean (ϵ ϵ vqk)-fuzzy generalized bi-ideals in ordered semigroups, which is a generalization of the concept of an (alpha, beta)-fuzzy generalized bi-ideal in an ordered semi-group. We also define (is an element of, is an element of boolean (ϵ ϵ vqk)-fuzzy left (resp. right)ideals. Using these concept, some characterization theorems of regular, left (resp. right) regular, completely regular and weakly regular ordered semigroups are provided. The upper/lower parts of an (is an element of, is an element of boolean (ϵ ϵ vqk)-fuzzy generalized bi-ideal and (is an element of, is an element of boolean (ϵ ϵ vqk)-fuzzy left (resp. right)-ideal are given, and some characterizations are provided

Models Development for Single-Row Networks from Connected Graphs

In this paper, we present a collection of models for connected graphs mapping into single–row networks. The collection involves three specific models for perfect binary trees, trees and partially dense graphs, and three general models for connected graphs. These models are compared in terms of their structures, energy values, congestion and number of doglegs in the single–row transformation. The numerical experiments are run by each respective developed program. The transformation is necessary in applications such as in the assignment of telephone channels to caller–receiver pairs roaming in cells in a cellular network on real–time basis

Linear-time heuristic partitioning technique for mapping of connected graphs into single-row networks

In this paper, a model called graph partitioning and transformation model (GPTM) which transforms a connected graph into a single-row network is introduced. The transformation is necessary in applications such as in the assignment of telephone channels to caller-receiver pairs roaming in cells in a cellular network on real-time basis. A connected graph is then transformed into its corresponding single-row network for assigning the channels to the caller-receiver pairs. The GPTM starts with the linear-time heuristic graph partitioning to produce two subgraphs with higher densities. The optimal labeling for nodes are then formed based on the simulated annealing technique. Experimental results support our hypothesis that GPTM efficiently transforms the connected graph into its single-row network