Commuting Graphs Of Dihedral Type Groups ∗ (original) (raw)

Commuting Graphs on Dihedral Group

Let Γ be a non-abelian group and Ω ⊆ Γ. The commuting graph C(Γ, Ω), has Ω as its vertex set with two distinct elements of Ω joined by an edge when they commute in Γ. In this paper we discuss certain properties of commuting graphs constructured on the dihedral group D 2n with respect to some specific subsets. More specifically we obtain the chromatic number and clique number of these commuting graphs. 2000 Mathematics Sub ject Classification: 05C.

The Commuting Graphs On Groups D2n And Qn

Journal of Mathematics and Computer Science

Given group , the commuting graph of , is defined as the graph with vertex set , and two distinct vertices and are connected by an edge, whenever they commute, that is. In this paper we get some parameters of graph theory, as independent number and clique number for groups , .

On the non-commuting graph of dihedral group

Electronic Journal of Graph Theory and Applications

For a nonabelian group G, the non-commuting graph Γ G of G is defined as the graph with vertexset G−Z(G), where Z(G) is the center of G, and two distinct vertices of Γ G are adjacent if they do not commute in G. In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of the non-commuting graph on D 2n. We also find the mean distance of the non-commuting graph on D 2n .

The commuting graphs on groups and

Given group, the commuting graph of, is defined as the graph with vertex set, and two distinct vertices and are connected by an edge, whenever they commute, that is. In this paper we get some parameters of graph theory, as independent number and clique number for groups,. Keywords: independent number, clique number, generalized quaternion group

A Kind of Non-commuting Graph of Finite Groups

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] ≠ g and [y,x] ≠ g. We denote this graph by g G Γ. In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number and dominating number. We prove that if G, H are isoclinic groups with |Z (G)|=|Z (H)|, then their associated graphs are isomorphic.

Some Meta-Cayley Graphs on Dihedral Groups

Graphs and Combinatorics, 2019

In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups.

The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

Proyecciones (Antofagasta), 2021

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

Some Properties of Coprime Graph of Dihedral Group D_2n When n is a Prime Power

2020

The Study of algebraic structures, especially on graphs theory, leads to anew topics of research in recent years. In this paper, the algebraic structures that will be represented by a coprime graph are the dihedral group and its subgroups. The coprime graph of a group G, denoted by \Gamma_D_2n is a graph whose vertices are elements of G and two distinct vertices a and b are adjacent if only if (|a,|b|)=1. Some properties of the coprime graph of a dihedral group D_2n are obtained. One of the results is if n is prime then \Gamma_D_2n is a complete bipartite graph. Moreover, if n is the power of prime then \Gamma_D_2n is a multipartite graph.

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

2010

Let G be a non-abelian group. The non-commuting graph A G 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 a complete subgraph of Γ is called the clique number of Γ and it is denoted by ω(Γ). In this paper we characterize all non-solvable groups G with ω(A G) ≤ 57, where the number 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2, 7). We also complete the determination of ω(A G) for all finite minimal simple groups. 1991 Mathematics Subject Classification. 20D60.

Full automorphism group of commuting and non-commuting graph of dihedral and symmetric groups

2018

An automorphism of finite graph G is a permutation on its vertex set that conserves adjacency. The set of all automorphism of G is a group under composition of function. This group is called the full automorphism group of G. Study on the full automorphism group is an interesting topic because most graphs have only the trivial automorphism and many special graphs have many automorphisms. One of the special graphs is graph that associated with group. The result of this paper is the full automorphism groups of commuting and non-commuting graph of non-abelian finite group, especially on dihedral and symmetric groups, according to the choice of their subgroups.