A Generalization of Noncommuting Graph via Automorphisms of a Group

2013 ◽  
Vol 42 (1) ◽  
pp. 174-185
Author(s):  
Z. Barati ◽  
A. Erfanian ◽  
K. Khashyarmanesh ◽  
Kh. Nafar
Keyword(s):  
Noncommuting Graph

scholarly journals Two bounds on the noncommuting graph

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Stefano Nardulli ◽  
Francesco G. Russo
Keyword(s):  
Finite Group ◽  
General Context ◽  
Sobolev Inequalities ◽  
Classical Analysis ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
A Finite Group ◽  
Noncommuting Graph ◽  
First Time

AbstractErdős introduced the noncommuting graph in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph has become relevant during the last years for various reasons. Here we deal with a numerical aspect, showing for the first time an isoperimetric inequality and an analytic condition in terms of Sobolev inequalities. This last result holds in the more general context of weighted locally finite graphs.

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).

A New Characterization ofA10by its Noncommuting Graph

2008 ◽  
Vol 36 (2) ◽  
pp. 523-528 ◽  
Author(s):  
Lingli Wang ◽  
Wujie Shi
Keyword(s):  
Noncommuting Graph

Recognition by noncommuting graph of finite simple groups L 4(q)

2010 ◽  
Vol 6 (1) ◽  
pp. 1-16 ◽  
Author(s):  
M. Akbari ◽  
M. Kheirabadi ◽  
A. R. Moghaddamfar
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Noncommuting Graph

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.

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.

On the noncommuting graph associated with a finite group

2005 ◽  
Vol 46 (2) ◽  
pp. 325-332 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
W. J. Shi ◽  
W. Zhou ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
A Finite Group ◽  
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.

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