scholarly journals The third-noncommuting graph of a group

2016 ◽  
Vol 34 (1) ◽  
pp. 279-284
Author(s):  
Maysam Zallaghi ◽  
Ali Iranmanesh
Keyword(s):  
Proper Subgroup ◽  
The Third ◽  
Vertex Set ◽  
Noncommuting Graph

‎Let $ G $ be a group and let $T^{3}(G)$ be the proper subgroup $\lbrace h\in G \vert (gh)^{3}=(hg)^{3},~for~all‎~ ‎g\in G\rbrace $ of $ G $‎. ‎\textit{The third-noncommuting graph} of $ G $ is the graph with‎ vertex set $ G\setminus T^{3}(G) $‎, ‎where two vertices $ x $ and $ y $ are adjacent if $ (xy)^{3}\neq (yx)^{3} $‎. In this paper‎, ‎at first we obtain some results for this graph for any group $G$‎. ‎Then‎, ‎we investigate the structure of this graph for some groups‎.

scholarly journals On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3147
Author(s):  
Monalisha Sharma ◽  
Rajat Kanti Nath ◽  
Yilun Shang
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Dihedral Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Noncommuting Graph

Let H be a subgroup of a finite non-abelian group G and g∈G. Let Z(H,G)={x∈H:xy=yx,∀y∈G}. We introduce the graph ΔH,Gg whose vertex set is G\Z(H,G) and two distinct vertices x and y are adjacent if x∈H or y∈H and [x,y]≠g,g−1, where [x,y]=x−1y−1xy. In this paper, we determine whether ΔH,Gg is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH

2009 ◽  
Vol 08 (01) ◽  
pp. 105-114 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Nonabelian Simple Group ◽  
Prime Graphs ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
A Finite Group ◽  
Noncommuting Graph ◽  
Almost All

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

scholarly journals Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

2018 ◽  
Vol 36 (2) ◽  
pp. 9-15
Author(s):  
Vida Ahmadi ◽  
Mohammad Reza Darafshe
Keyword(s):  
Connected Graph ◽  
Zagreb Indices ◽  
The Third ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Simple Connected Graph

Let  be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: ,   and   where  is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph  are defind by:  and . The first and second Zagreb coindices of graph are defind by:  and .  and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some  graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

scholarly journals The General Connectivity and General Sum-Connectivity Indices of Nanostructures

2015 ◽  
Vol 44 ◽  
pp. 73-80
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Quantitative Structure Activity Relationship ◽  
Connectivity Index ◽  
Quantitative Structure ◽  
Structure Property ◽  
The Third ◽  
Structure Activity ◽  
New Variant ◽  
Connectivity Indices ◽  
Structure Property Relationship ◽  
Vertex Set

Let G be a simple graph with vertex set V(G) and edge set E(G). For ∀νi∈V(G),di denotes the degree of νi in G. The Randić connectivity index of the graph G is defined as [1-3] χ(G)=∑e=v1v2є(G)(d1d2)-1/2. The sum-connectivity index is defined as χ(G)=∑e=v1v2є(G)(d1+d2)-1/2. The sum-connectivity index is a new variant of the famous Randić connectivity index usable in quantitative structure-property relationship and quantitative structure-activity relationship studies. The general m-connectivety and general m-sum connectivity indices of G are defined as mχ(G)=∑e=v1v2...vim+1(1/√(di1di2...dim+1)) and mχ(G)=∑e=v1v2...vim+1(1/√(di1+di2+...+dim+1)) where vi1vi2...vim+1 runs over all paths of length m in G. In this paper, we introduce a closed formula of the third-connectivity index and third-sum-connectivity index of nanostructure "Armchair Polyhex Nanotubes TUAC6[m,n]" (m,n≥1).

RECOGNITION OF THE PROJECTIVE GENERAL LINEAR GROUP PGL(2, q) BY ITS NONCOMMUTING GRAPH

2011 ◽  
Vol 10 (02) ◽  
pp. 201-218 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Linear Group ◽  
General Linear Group ◽  
Simple Groups ◽  
General Linear ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Vertex Set ◽  
Noncommuting Graph ◽  
Projective General Linear Group

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL (2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.

scholarly journals The independence polynomial of conjugate graph and noncommuting graph of groups of small order.

2017 ◽  
Vol 13 (4) ◽  
pp. 602-605
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Independent Set ◽  
Graph Of Groups ◽  
Independence Polynomial ◽  
Central Elements ◽  
Vertex Set ◽  
Noncommuting Graph ◽  
Nonabelian Groups

An independent set of a graph is a set of pairwise non-adjacent vertices. 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 conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate. The noncommuting graph is defined as a graph whose vertex set is non-central elements in which two vertices are adjacent if and only if they do not commute. In this research, the independence polynomial of the conjugate graph and noncommuting graph are found for three nonabelian groups of order at most eight.

scholarly journals On r-Noncommuting Graph of Finite Rings

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 233
Author(s):  
Rajat Kanti Nath ◽  
Monalisha Sharma ◽  
Parama Dutta ◽  
Yilun Shang
Keyword(s):  
Undirected Graph ◽  
Clique Number ◽  
Star Graph ◽  
Finite Rings ◽  
Induced Subgraph ◽  
Noncommutative Rings ◽  
Central Elements ◽  
Vertex Set ◽  
Vertex Degrees ◽  
Noncommuting Graph

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

RECOGNIZING SOME FINITE SIMPLE GROUPS BY NONCOMMUTING GRAPH

2012 ◽  
Vol 11 (04) ◽  
pp. 1250077 ◽  
Author(s):  
M. KHEIRABADI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Linear Groups ◽  
Nonabelian Group ◽  
Projective Special Linear Groups ◽  
Prime Graphs ◽  
Vertex Set ◽  
Trivial Center ◽  
Noncommuting Graph

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.

scholarly journals ON THE PRIME GRAPH OF SIMPLE GROUPS

2014 ◽  
Vol 91 (2) ◽  
pp. 227-240 ◽  
Author(s):  
TIMOTHY C. BURNESS ◽  
ELISA COVATO
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Prime Graphs ◽  
Vertex Set ◽  
A Finite Group

AbstractLet $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

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