scholarly journals The Non-Commuting, Non-Generating Graph of a Nilpotent Group

10.37236/9802 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Peter Cameron ◽  
Saul Freedman ◽  
Colva Roney-Dougal
Keyword(s):  
Maximal Subgroup ◽  
Nilpotent Group ◽  
Commuting Graph ◽  
Generating Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Infinite Case ◽  
The Difference ◽  
The Relationship

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

THE EXISTENCE OR NONEXISTENCE OF NON-COMMUTING GRAPHS WITH PARTICULAR PROPERTIES

2013 ◽  
Vol 13 (01) ◽  
pp. 1350064 ◽  
Author(s):  
M. AKBARI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Strongly Regular Graph ◽  
Complete Multipartite Graph ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Strongly Regular ◽  
Multipartite Graph ◽  
Complete Multipartite

We consider the non-commuting graph ∇(G) of a non-abelian finite group G; its vertex set is G\Z(G), the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if [x, y] ≠ 1. We determine the structure of any finite non-abelian group G (up to isomorphism) for which ∇(G) is a complete multipartite graph (see Propositions 3 and 4). It is also shown that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph. Finally, it is proved that there is no non-abelian group whose non-commuting graph is self-complementary and n-cube.

scholarly journals The independence polynomial of inverse commuting graph of dihedral groups

2020 ◽  
Vol 16 (1) ◽  
pp. 115-120
Author(s):  
Aliyu Suleiman ◽  
Aliyu Ibrahim Kiri
Keyword(s):  
Identity Element ◽  
Independent Set ◽  
Independent Sets ◽  
Dihedral Groups ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Set of vertices not joined by an edge in a graph is called the independent set of the graph. The independence polynomial of a graph is a polynomial whose coefficient is the number of independent sets in the graph. In this research, we introduce and investigate the inverse commuting graph of dihedral groups (D2N) denoted by GIC. It is a graph whose vertex set consists of the non-central elements of the group and for distinct  x,y, E D2N, x and y are adjacent if and only if xy = yx = 1  where 1 is the identity element. The independence polynomials of the inverse commuting graph for dihedral groups are also computed. A formula for obtaining such polynomials without getting the independent sets is also found, which was used to compute for dihedral groups of order 18 up to 32.

On the non-commuting graph in finite Moufang loops

2018 ◽  
Vol 17 (04) ◽  
pp. 1850070
Author(s):  
Karim Ahmadidelir
Keyword(s):  
Abelian Group ◽  
Nilpotent Groups ◽  
Moufang Loop ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Commutative Moufang Loop ◽  
Commuting Graph ◽  
Moufang Loops ◽  
Central Elements ◽  
Vertex Set

The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

scholarly journals The Topological Indices of the Non-commuting Graph for Symmetric Groups

2020 ◽  
pp. 1-5
Author(s):  
Nur Idayu Alimon ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Symmetric Group ◽  
Chemical Compound ◽  
Molecular Graph ◽  
Symmetric Groups ◽  
Wiener Index ◽  
Topological Indices ◽  
Zagreb Index ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

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 S_n, 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. Keywords: Wiener index; Zagreb index; non-commuting graph; symmetric groups

On the Clique Numbers of Non-commuting Graphs of Certain Groups

2010 ◽  
Vol 17 (04) ◽  
pp. 611-620 ◽  
Author(s):  
A. Abdollahi ◽  
A. Azad ◽  
A. Mohammadi Hassanabadi ◽  
M. Zarrin
Keyword(s):  
Solvable Groups ◽  
Maximum Size ◽  
Clique Number ◽  
Special Linear Group ◽  
Simple Graph ◽  
Simple Groups ◽  
Projective Special Linear Group ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.

scholarly journals On the generalized commuting and non-commuting graph for metacyclic 3-groups

2017 ◽  
Vol 13 (3) ◽  
Author(s):  
Siti Norziahidayu Amzee Zamri ◽  
Nor Haniza Sarmin ◽  
Mustafa Anis El-Sanfaz ◽  
Hamisan Rahmat
Keyword(s):  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Empty Subset

Let   be a metacyclic 3-group and let   be a non-empty subset of   such that  . The generalized commuting and non-commuting graphs of a group   is denoted by   and   respectively. The vertex set of the generalized commuting and non-commuting graphs are the non-central elements in the set   such that     where   Two vertices in   are joined by an edge if they commute, meanwhile, the vertices in   are joined by an edge if they do not commute.

Comparison of In-the-Ear and Over-the-Ear Hearing Aid Fittings

1986 ◽  
Vol 51 (4) ◽  
pp. 362-369 ◽  
Author(s):  
Donna M. Risberg ◽  
Robyn M. Cox
Keyword(s):  
Hearing Aids ◽  
High Frequency ◽  
Comparative Evaluation ◽  
Speech Intelligibility ◽  
Hearing Aid ◽  
Hearing Aid Fitting ◽  
The Difference ◽  
Functional Gain ◽  
The Relationship

A custom in-the-ear (ITE) hearing aid fitting was compared to two over-the-ear (OTE) hearing aid fittings for each of 9 subjects with mild to moderately severe hearing losses. Speech intelligibility via the three instruments was compared using the Speech Intelligibility Rating (SIR) test. The relationship between functional gain and coupler gain was compared for the ITE and the higher rated OTE instruments. The difference in input received at the microphone locations of the two types of hearing aids was measured for 10 different subjects and compared to the functional gain data. It was concluded that (a) for persons with mild to moderately severe hearing losses, appropriately adjusted custom ITE fittings typically yield speech intelligibility that is equal to the better OTE fitting identified in a comparative evaluation; and (b) gain prescriptions for ITE hearing aids should be adjusted to account for the high-frequency emphasis associated with in-the-concha microphone placement.

Relationship between total plasma homocysteine and the risk of aneurysms – a meta-analysis

VASA ◽  
2020 ◽  
pp. 1-6
Author(s):  
Hanji Zhang ◽  
Dexin Yin ◽  
Yue Zhao ◽  
Yezhou Li ◽  
Dejiang Yao ◽  
...  
Keyword(s):  
Large Scale ◽  
Meta Analysis ◽  
Single Species ◽  
Total Plasma ◽  
Control Groups ◽  
Randomized Controlled ◽  
Randomized Controlled Studies ◽  
The Difference ◽  
The Relationship ◽  
Healthy Participants

Summary: Our meta-analysis focused on the relationship between homocysteine (Hcy) level and the incidence of aneurysms and looked at the relationship between smoking, hypertension and aneurysms. A systematic literature search of Pubmed, Web of Science, and Embase databases (up to March 31, 2020) resulted in the identification of 19 studies, including 2,629 aneurysm patients and 6,497 healthy participants. Combined analysis of the included studies showed that number of smoking, hypertension and hyperhomocysteinemia (HHcy) in aneurysm patients was higher than that in the control groups, and the total plasma Hcy level in aneurysm patients was also higher. These findings suggest that smoking, hypertension and HHcy may be risk factors for the development and progression of aneurysms. Although the heterogeneity of meta-analysis was significant, it was found that the heterogeneity might come from the difference between race and disease species through subgroup analysis. Large-scale randomized controlled studies of single species and single disease species are needed in the future to supplement the accuracy of the results.

Klein with Lacan: A Study on the Reception of Lacanian Ideas in Uruguay and Its Effects on Clinical Practices (1955–82)

2020 ◽  
Vol 22 (3) ◽  
pp. 341-361
Author(s):  
Gonzalo Grau-Pérez ◽  
J. Guillermo Milán
Keyword(s):  
Clinical Practice ◽  
Clinical Material ◽  
Clinical Practices ◽  
Before And After ◽  
The Difference ◽  
The 1960S ◽  
The Relationship ◽  
Discursive Approach ◽  
The Way ◽  
Clinical Cases

In Uruguay, Lacanian ideas arrived in the 1960s, into a context of Kleinian hegemony. Adopting a discursive approach, this study researched the initial reception of these ideas and its effects on clinical practices. We gathered a corpus of discursive data from clinical cases and theoretical-doctrinal articles (from the 1960s, 1970s and 1980s). In order to examine the effects of Lacanian ideas, we analysed the difference in the way of interpreting the clinical material before and after Lacan's reception. The results of this research illuminate some epistemological problems of psychoanalysis, especially the relationship between theory and clinical practice.

The End of the Word as We Know It

2012 ◽  
Vol 6 (1-3) ◽  
pp. 165-184
Author(s):  
Timothy Beal
Keyword(s):  
Cultural Meanings ◽  
Cultural Icon ◽  
The Bible ◽  
Material Objectivity ◽  
The Difference ◽  
The Relationship

This essay attends to a distinction that requires closer examination and theorization in our discourse on iconic books and other scriptures: the difference between iconic object and cultural icon. How do we conceive of relations between the particular, ritualized iconicities of particular scriptures in particular religious contexts and the cultural iconicities of scriptures in general, such as “the Bible” or “the Quran,” whose visual and material objectivity is highly ambiguous? How if at all are the iconic cultural meanings of the ideas of such books related to the particular iconic textual objects more or less instantiate them? These questions are explored through particular focus on the relationship between the particular iconicities of particular print Bibles, as iconic objects, and the general iconicity of the cultural icon of the Bible.

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