Finite groups with star-free noncyclic graphs

The graph related to conjugacy classes is a graph whose vertices are non central conjugacy classes, where two vertices are connected by an edge if they are not coprime. In this paper, we generalize the graph related to conjugacy classes by defining a graph called generalized conjugacy classes graph. This graph is found for some groups. Besides, some graph properties are provided.

Siberian Mathematical Journal, 2005

Let G be a finite group. We define the noncommuting graph ∇(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. We study some properties of ∇(G) and prove that, for many groups G, if H is a group with ∇(G) isomorphic to ∇(H) then |G| = |H|.

Let G be a finite non-abelian group and let Ω be a set of elements of G. Let A be the set of commuting elements in , i.e A = {v ∈ Ω : vg = gv, g ∈ G}. In this paper, we extend the work on conjugate graph by defining a new graph called the orbit graph, denoted as Γ^{Ω}_G. The vertices of Γ^{Ω}_G are non- central elements in Ω but not in A in which two vertices of Γ^{Ω}_G are adjacent if they are conjugate. Some graph properties are provided. Besides, the orbit graph of dihedral groups and quaternion groups is determined.

In this paper we classify the finite groups satisfying the following property P 4 : their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.

European Journal of Pure and Applied Mathematics, 2022

The interplay of groups and graphs has been a subject of interest by mathematics researchers nowadays. One particular instance is the identity graph of a group introduced by Kandasamy [6]. Moreover, the concept of a central graph of any graph is widely used by many graph theorist. The central graph of a graph G denoted by C(G) can be obtained by subdividing the edge of G exactly once and joining all the nonadjacent vertices of G in C(G). In this paper, we construct the central graph of the identity graph of finite cyclic group and investigate some of its graph properties.

Suppose n is a fixed positive integer. We introduce the relative n-th non-commuting graph Γ n H,G , associated to the nonabelian subgroup H of group G. The vertex set is G \ C n H,G in which C n H,G = {x ∈ G : [x, y n ] = 1 and [x n , y] = 1 for all y ∈ H}. Moreover, {x, y} is an edge if x or y belong to H and xy n = y n x or x n y = yx n. In fact, the relative n-th commutativity degree, Pn(H, G) the probability that n-th power of an element of the subgroup H commutes with another random element of the group G and the non-commuting graph are the keys to construct such a graph. It is proved that two isoclinic non-abelian groups have isomorphic graphs under special conditions.

Mathematics

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.

The non-cyclic graph C G to a non locally cyclic group G is as follows: take G \ C yc(G) as vertex set, where C yc(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. In this paper, we classify the finite groups whose non-cyclic graphs are outerplanar. Also all finite groups whose non-cyclic graphs can be embedded on the torus or projective plane are classified.

Note di Matematica, 2018

The nilpotent conjugacy class graph (or NCC-graph) of a group G is a graph whose vertices are the nontrivial conjugacy classes of G such that two distinct vertices x G and y G are adjacent if x , y is nilpotent for some x ∈ x G and y ∈ y G. We discuss on the number of connected components as well as diameter of connected components of these graphs. Also, we consider the induced subgraph Γn(G) of the NCC-graph with vertices set {g G | g ∈ G \ Nil(G)}, where Nil(G) = {g ∈ G | x, g is nilpotent for all x ∈ G}, and classify all finite non-nilpotent group G with empty and triangle-free NCC-graphs.

Archiv der Mathematik, 2005