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 NON-CYCLIC GRAPH ASSOCIATED WITH A GROUP

2009 ◽  
Vol 08 (02) ◽  
pp. 243-257 ◽  
Author(s):  
A. ABDOLLAHI ◽  
A. MOHAMMADI HASSANABADI
Keyword(s):  
Solvable Group ◽  
Cyclic Group ◽  
Cyclic Subgroup ◽  
Maximum Size ◽  
Clique Number ◽  
Simple Graph ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Cyclic Graphs ◽  
Cyclic Graph

We associate a graph [Formula: see text] to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\ Cyc (G) as vertex set, where Cyc (G) = {x ∈ G | 〈x,y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever [Formula: see text] and the equality for a non-solvable group G holds if and only if G/ Cyc (G) ≅ A5 or S5.

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.

Non-commuting graphs of certain almost simple groups

2019 ◽  
Vol 12 (05) ◽  
pp. 1950081
Author(s):  
M. Jahandideh ◽  
R. Modabernia ◽  
S. Shokrolahi
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Commuting Graph ◽  
Vertex Set

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

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.

scholarly journals On Generalized Non-commuting Graph of a Finite Ring

2018 ◽  
Vol 25 (01) ◽  
pp. 149-160 ◽  
Author(s):  
Jutirekha Dutta ◽  
Dhiren K. Basnet ◽  
Rajat K. Nath
Keyword(s):  
Simple Graph ◽  
Finite Ring ◽  
Dominating Sets ◽  
Finite Rings ◽  
Commuting Graph ◽  
Vertex Set

Let S and K be two subrings of a finite ring R. Then the generalized non-commuting graph of subrings S, K of R, denoted by ГS,K, is a simple graph whose vertex set is [Formula: see text], and where two distinct vertices a, b are adjacent if and only if [Formula: see text] or [Formula: see text] and [Formula: see text]. We determine the diameter, girth and some dominating sets for ГS,K. Some connections between ГS,K and Pr(S, K) are also obtained. Further, ℤ-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two ℤ-isoclinic pairs are isomorphic under some conditions.

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.

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

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.

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