Non-commuting Graph Research Articles

A Generalization of Noncommuting Graph via Automorphisms of a Group

Let G be a group. By using a family 𝒜 of subsets of automorphisms of G, we introduced a simple graph Γ𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ𝒜(G).

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

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.

Simple groups are characterized by their non-commuting graphs

Abstract. The non-commuting graph of a finite group G is a highly symmetrical object (indeed, embeds in ), yet its complexity pales in comparison to that of G. Still, it is natural to seek conditions under which G can be reconstructed from . Surely some conditions are necessary, as is evidenced by the minuscule example . A conjecture made in [J. Algebra 298 (2006), 468–492], commonly referred to as the AAM Conjecture, proposes that the property of being a nonabelian simple group is sufficient. In [Sib. Math. J. 49 (2008), no. 6, 1138–1146], this conjecture is verified for all sporadic simple groups, while in [J. Algebra 357 (2012), 203–207], it is verified for the alternating groups. In this paper we verify it for the simple groups of Lie type, thereby completing the proof of the conjecture.

Characterization of the symmetric group by its non-commuting graph

‎The non-commuting graph $nabla(G)$ of a non-abelian group $G$ is defined as‎ ‎follows‎: ‎its vertex set is $G-Z(G)$ and two distinct vertices $x$ and $y$ are‎ ‎joined by an edge if and only if the commutator of $x$ and $y$ is not the‎ ‎identity‎. ‎In this paper we prove that if $G$ is a finite group with‎ ‎$nabla(G) cong nabla(BS_n)$‎, ‎then $G cong BS_n$‎, ‎where $BS_n$‎ ‎is the symmetric group of degree $n$‎, ‎where $n$ is a natural number‎.

Commutativity Pattern of Finite Non-Abelian p-Groups Determine Their Orders

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ G (called noncommuting graph of G) with G as follows: Take G∖Z(G) as the vertices of Γ G , and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that “the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that Γ P ≅ Γ H for some group H, then |P| = |H|.

A CHARACTERIZATION OF THE GROUP A22BY NON-COMMUTING GRAPH

Let G be a finite non-abelian group. We define the non-commuting graph <TEX>${\nabla}(G)$</TEX> of G as follows: the vertex set of <TEX>${\nabla}(G)$</TEX> is G-Z(G) and two vertices x and y are adjacent if and only if <TEX>$xy{\neq}yx$</TEX>. In this paper we prove that if G is a finite group with <TEX>$${\nabla}(G){\simeq_-}{\nabla}(\mathbb{A}_{22})$$</TEX>, then <TEX>$$G{\simeq_-}\mathbb{A}_{22}$$</TEX>where <TEX>$\mathbb{A}_{22}$</TEX> is the alternating group of degree 22.

RELATIVE NON-COMMUTING GRAPH OF A FINITE GROUP

The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

RECOGNIZING SOME FINITE SIMPLE GROUPS BY 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.

Some results on characterization of finite group by non commuting graph

The non commuting graph r(G) of a non-abelian nite group G is dened as follows: its vertex set is G Z(G) and two distinct vertices x and y are joined by an edge if and only if the commutator of x and y is not the identity. In this paper we prove some new results about this graph. In particular we will give a new proof of Theorem 3.24 of (A. Abdollahi, S. Akbari, H. R, Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006) 468-492.). We also prove that if

The non-nilpotent graph of a semigroup

We associate a graph \({\mathcal{N}}_{S}\) with a semigroup S (called the upper non-nilpotent graph of S). The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case S is a group this graph has been introduced by A. Abdollahi and M. Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of S determined by the associated graph. It is shown that if a finite semigroup S has empty upper non-nilpotent graph then S is positively Engel. On the other hand, a semigroup has a complete upper non-nilpotent graph if and only if it is a completely simple semigroup that is a band. One of the main results states that if all connected \({\mathcal{N}}_{S}\)-components of a semigroup S are complete (with at least two elements) then S is a band that is a semilattice of its connected components and, moreover, S is an iterated total ideal extension of its connected components. We also show that some graphs, such as a cycle C n on n vertices (with n≥5), are not the upper non-nilpotent graph of a semigroup. Also, there is precisely one graph on 4 vertices that is not the upper non-nilpotent graph of a semigroup with 4 elements. This work also is a continuation of earlier work by Okniński, Riley and the first named author on (Malcev) nilpotent semigroups.

Characterization of the alternating group by its non-commuting graph

Let G be any non-abelian group and Z(G) be its center. The non-commuting graph ΓG of G is the simple graph whose vertex set is G∖Z(G), with two vertices x and y adjacent whenever xy≠xy. We prove that if ΓG is isomorphic to the non-commuting graph of the alternating group An (n⩾4), then G≅An. This result together with a recent one due to Solomon and Woldar gives a complete positive answer to a conjecture proposed in [A. Abdollahi, S. Akbari, H.R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006) 468–492]: If S is any finite non-abelian simple group such that ΓS≅ΓG for some group G, then G≅S.

On the relation between the non-commuting graph and the prime graph

‎$pi(G)$ denote the set of prime divisors of the order of $G$ and‎ ‎denote by $Z(G)$ the center of $G$‎. ‎Thetextit{ prime graph} of‎ ‎$G$ is the graph with‎ ‎vertex set $pi(G)$ where two distinct primes $p$ and $q$ are‎ ‎joined by an edge if and only if $G$‎ ‎contains an element of order $pq$ and the textit{non-commuting‎ ‎graph} of $G$ is the graph with the vertex set $G-Z(G)$ where two‎ ‎non-central elements $x$ and $y$ are‎ ‎joined by an edge if and only if $xy neq yx$‎. ‎Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs‎. ‎In this article‎, ‎we show that if $ | Z ( G ) | = | Z ( H ) | $‎, ‎then‎ ‎$ G $ and $ H $ have the same prime graphs and also‎, ‎the set of‎ ‎orders of the maximal abelian subgroups of $ G $ and $ H $ are‎ ‎the same‎.

Problems of Connectivity between the Sylow Graph,the Prime Graph and the Non-Commuting Graph of a Group

The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid CFSG or not. On the other hand, what happens for infinite groups? Since the status of knowledge of the non-commuting graph and of the prime graph is satisfactory, is it possible to find relations between these two graphs and the Sylow graph? In the present note we make the point of the situation and formulate the above questions in appropriate way.

Characterization of Aut(J2) and Aut(McL) by Their Non-commuting Graphs

For a non-abelian group G, we associate the non-commuting graph ∇ (G) whose vertex set 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 this paper, we prove that Aut (J2) and Aut (McL) are characterized by their non-commuting graphs.

Abelian coverings of finite general linear groups and an application to their non-commuting graphs

In this paper we introduce and study a family \(\mathcal{A}_{n}(q)\) of abelian subgroups of \({\rm GL}_{n}(q)\) covering every element of \({\rm GL}_{n}(q)\). We show that \(\mathcal{A}_{n}(q)\) contains all the centralizers of cyclic matrices and equality holds if q>n. For q>2, we obtain an infinite product expression for a probabilistic generating function for \(|\mathcal{A}_{n}(q)|\). This leads to upper and lower bounds which show in particular that $$c_1q^{-n}\leq \frac{|\mathcal{A}_n(q)|}{|\mathrm{GL}_n(q)|}\leq c_2q^{-n}$$ for explicit positive constants c1,c2. We also prove that similar upper and lower bounds hold for q=2.

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

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.

Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph

Let G be a finite non-Abelian group. We define a graph Γ G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ S ≅ Γ G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except $ {\mathbb{A}_{10}} $ , L 4(8), L 4(4), and U 4(4). In this paper, we prove that if $ {\mathbb{A}_{16}} $ denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism $ {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} $ implies that $ {\mathbb{A}_{16}} \cong G $ .

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

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.

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

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x, y] ≠ 1. We prove that if L ∈ {L4(7), L4(11), L4(13), L4(16), L4(17)} and G is a finite group such that ∇(G) ≅ ∇(L), then G ≅ L.

Noncommuting graphs of matrices over semirings

We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, all noninvertible matrices over a semiring, and the full matrix semiring. In nearly all cases we prove that diameters are less than or equal to 2 and girths are less than or equal to 3, except in the case of 2 × 2 nilpotent matrices.