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

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.

On r-Noncommuting Graph of Finite Rings

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.

Connectivity and planarity of g-noncommuting graph of finite groups

Let [Formula: see text] be a finite non-abelian group and [Formula: see text] be its center. For a fixed nonidentity element [Formula: see text] of [Formula: see text], the [Formula: see text]-noncommuting graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph in which its vertices are [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we discuss about connectivity of [Formula: see text] and determine all finite non-abelian groups such that their [Formula: see text]-noncommuting graphs are 1-planar, toroidal or projective.

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

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.

The third-noncommuting graph of a group

‎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.

Bilangan Kromatik Grap Commuting dan Non Commuting Grup Dihedral

Commuting graph is a graph that has a set of points X and two different vertices to be connected directly if each commutative in G. Let G non abelian group and Z(G) is a center of G. Noncommuting graph is a graph which the the vertex is a set of G\Z(G) and two vertices x and y are adjacent if and only if xy≠yx. The vertex colouring of G is giving k colour at the vertex, two vertices that are adjacent not given the same colour. Edge colouring of G is two edges that have common vertex are coloured with different colour. The smallest number k so that a graph can be coloured by assigning k colours to the vertex and edge called chromatic number. In this article, it is available the general formula of chromatic number of commuting and noncommuting graph of dihedral group

Two bounds on the noncommuting graph

Erdős 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.

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.

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.