Non-commuting Graph Research Articles

Szeged and Padmakar-Ivan Energies of Non-Commuting Graph for Dihedral Groups

The graph can represent the molecule structure and the -electron energy derives the concept of graph energy. The graph also can be related to the groups or rings as its vertex set. The non-commutative graph is a type of graph whose construction is determined by the structure of a group. This paper focuses on the energy of the non-commuting graph for dihedral groups using the Szeged and Padmakar-Ivan matrices. Both matrices are constructed based on the distance between two vertices in the graph. The eigenvalues ​​of these matrices lead to the formulation of the graph's energy values​. Interestingly, the energies obtained are equal to twice the spectral radius and are hyperenergetic.

On the Non-Commuting Graph of the Group U6n

Let G be a finite group. The non-commuting graph of G is a simple graph Γ(G) whose vertices are elements of G∖Z(G), where Z(G) is the center of G, and two distinct vertices a and b are joint by an edge if ab≠ba. In this paper, we study the non-commuting graph of the group U6n. The independent number, clique and chromatic numbers of the non-commuting graph of the group U6n, Γ(U6n), are determined. Additionally, the resolving polynomial, total eccentricity and independent polynomials of Γ(U6n) are computed. Finally, the detour and eccentric connectivity indices of Γ(U6n) are found.

Signless Laplacian energies of non-commuting graphs of finite groups and related results

The non-commuting graph of a non-abelian group [Formula: see text] with center [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] are adjacent if [Formula: see text]. In this paper, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of certain finite non-abelian groups. We obtain several conditions such that the non-commuting graph of [Formula: see text] is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the hyperenergetic and hypoenergetic properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic.

Sombor Index and Sombor Polynomial of the Noncommuting Graph Associated to Some Finite Groups

Sombor index is a newly developed degree-based topological index which involves the degree of the vertex in a simple connected graph. The Sombor index is known as the square root of the sum of the squared degrees of two adjacent vertices in a graph. Meanwhile, the noncommuting graph associated to a group is a graph where its vertices are the non-central elements of the group and two vertices are adjacent if and only if they do not commute. In this study, a new notion called the Sombor polynomial is introduced. Then, the general formula of the Sombor index and the Sombor polynomial of the noncommuting graph associated to some finite groups are determined by using their definitions and some preliminaries. The groups involved in this research are the dihedral groups, the quasidihedral groups, and the generalized quaternion groups. The results found can help the chemists and biologists to predict the chemical and physical properties of the molecules without involving any laboratory work.

Projective non-commuting graph of a group

Let $G$ be a finite non-abelian group and let $T$ be a transversal of the center of $G$ in $G$. The non-commuting graph of $G$ on a transversal of the center is the graph whose vertices are the non-central elements of $T$ and two vertices $x$ and $y$ are joined by an edge whenever $xy \neq yx$. In this paper, we classify the groups whose non-commuting graph on a transversal of the center is projective.

On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

The non-commuting graph, denoted by , is defined on a finite group , with its vertices are elements of excluding those in the center of . In this graph, two distinct vertices are adjacent whenever they do not commute in . The graph can be associated with several matrices including the most basic matrix, which is the adjacency matrix, , and a matrix called Sombor matrix, denoted by . The entries of are either the square root of the sum of the squares of degrees of two distinct adjacent vertices, or zero otherwise. Consequently, the adjacency and Sombor energies of is the sum of the absolute eigenvalues of the adjacency and Sombor matrices of , respectively, whereas the spectral radius of is the maximum absolute eigenvalues. Throughout this paper, we find the spectral radius obtained from the spectrum of and the Sombor energy of for dihedral groups of order , where . Moreover, there is an almost linear correlation between the Sombor energy and the adjacency energy of for which is slightly different than reported earlier in previous literature.

Closeness Energy of Non-Commuting Graph for Dihedral Groups

This paper focuses on the non-commuting graph for dihedral groups of order 2n, D2n, where n ≥ 3. We show the spectrum and energy of the graph corresponding to the closeness matrix. The result is that the obtained energy is always twice its spectral radius and is never an odd integer. Moreover, it is classified as hypoenergetic.

Wiener-Hosoya Energy of Non-Commuting Graph for Dihedral Groups

Wiener-Hosoya Energy of Non-Commuting Graph for Dihedral Groups

Finite groups with isomorphic non-commuting graphs have the same nilpotency property

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph ΓG associated to G is the graph whose vertex set is G∖Z(G) and two distinct elements x,y are adjacent if and only if xy≠yx. We prove that if G and H are non-abelian groups with isomorphic non-commuting graphs, such that G is nilpotent, then H is nilpotent, provided |Z(G)|≥|Z(H)|. Actually we prove conjecture 3 proposed in V. Grazian and C. Monetta (2023) [5].

Inequalities involving energy and Laplacian energy of non-commuting graphs of finite groups

Inequalities involving energy and Laplacian energy of non-commuting graphs of finite groups

Universal adjacency spectrum of the commuting and non-commuting graphs on AC-groups

For a simple undirected graph [Formula: see text] and [Formula: see text], [Formula: see text], the universal adjacency matrix [Formula: see text], where [Formula: see text] is the identity matrix, [Formula: see text] is the all-ones matrix, and [Formula: see text] and [Formula: see text] are the adjacency and degree diagonal matrices of [Formula: see text], respectively. For a finite non-abelian group [Formula: see text] and [Formula: see text], the commuting graph [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and the adjacency rule is commuting. The non-commuting graph of [Formula: see text] is the complement of [Formula: see text]. An AC-group is a non-abelian group such that the centralizer of every non-central element is abelian. In this paper, we obtain the universal adjacency eigenpairs of the commuting and non-commuting graphs for an arbitrary AC-group [Formula: see text] with [Formula: see text] or [Formula: see text]. Moreover, we determine the eigenpairs of [Formula: see text] when [Formula: see text] is a specific kind of group.

Harmonious colouring of line graph of commuting and non-commuting graph of D_{2n}

Harmonious coloring of graph \(G\) is a proper vertex coloring, where each pair of colors occurs at most on one pair of adjacent vertices. Minimum number of colors required for Harmonious coloring of \(G\) is the harmonious chromatic number, \(\chi_{H}(G)\). Here we determine the Harmonious chromatic number of the line graph of commuting graph and non-commuting graph of the dihedral group, \(D_{2n}\).

Generalization of Randic ́ Index of the Non-commuting Graph for Some Finite Groups

Randić index is one of the classical graph-based molecular structure descriptors in the field of mathematical chemistry. The Randić index of a graph is calculated by summing the reciprocals of the square root of the product of the degrees of two adjacent vertices in the graph. 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 and only if they do not commute. In this paper, the general formula of the Randić index of the non-commuting graph associated to three types of finite groups are presented. The groups involved are the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups. Some examples of the Randić index of the non-commuting graph related to a certain order of these groups are also given based on the main results.

A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if G and H are finite groups with isomorphic non-commuting graphs and G is nilpotent, then H must be nilpotent as well (Conjecture 2). We characterize the structure of such an H when G is a finite AC-group, that is, a finite group in which all centralizers of non-central elements are abelian. As an application, we prove Conjecture 2 for finite AC-groups whenever |Z(G)|≥|Z(H)|.

ON THE SZEGED INDEX AND ITS NON-COMMUTING GRAPH

In chemistry, the molecular structure can be represented as a graph. Based on the information from the graph, its characterization can be determined by computing the topological index. Topological index is a numerical value that can be computed by using some algorithms and properties of the graph. Meanwhile, the non-commuting graph is a graph, in which two distinct vertices are adjacent if and only if they do not commute, where it is made up of the non-central elements in a group as a vertex set. In this paper, the Szeged index of the non-commuting graph of some finite groups are computed. This paper focuses on three finite groups which are the quasidihedral groups, the dihedral groups, and the generalized quaternion groups. The construction of the graph is done by using Maple software. In finding the Szeged index, some of the previous results and properties of the graph for the quasidihedral groups, the dihedral groups, and the generalized quaternion groups are used. The generalisation of the Szeged index of the non-commuting graph is then established for the aforementioned groups. The results are then applied to find the Szeged index of the non-commuting graph of ammonia molecule.

Neighbors Degree Sum Energy of Commuting and Non-Commuting Graphs for Dihedral Groups

The neighbors degree sum (NDS) energy of a graph is determined by the sum of its absolute eigenvalues from its corresponding neighbors degree sum matrix. The non-diagonal entries of NDS−matrix are the summation of the degree of two adjacent vertices, or it is zero for non-adjacent vertices, whereas for the diagonal entries are the negative of the square of vertex degree. This study presents the formulas of neighbors degree sum energies of commuting and non-commuting graphs for dihedral groups of order 2n, D2n, for two cases−odd and even n. The results in this paper comply with the well known fact that energy of a graph is neither an odd integer nor a square root of an odd integer.

On the spectra of non-commuting graphs of certain class of groups

Let [Formula: see text] be the center of a finite non-abelian group [Formula: see text] The non-commuting graph of [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] A group [Formula: see text] is called a CA-group if the centralizer [Formula: see text] for every non central element [Formula: see text] is abelian. In this paper, we investigate the distance, distance (signless) Laplacian spectra of non-commuting graphs of some CA-groups, and obtain some conditions on them so that the corresponding non-commuting graph is distance, distance (signless) Laplacian integral.

DEGREE SUM ENERGY OF NON-COMMUTING GRAPH FOR DIHEDRAL GROUPS

For a finite group G, let Z(G) be the centre of G. Then the non-commuting graph on G, denoted by ΓG, has G\Z(G) as its vertex set with two distinct vertices vp and vq joined by an edge whenever vpvq ≠ vqvp. The degree sum matrix of a graph is a square matrix whose (p,q)-th entry is dvp + dvq whenever p is different from q, otherwise, it is zero, where dvi is the degree of the vertex vi. This study presents the general formula for the degree sum energy, EDS (ΓG), for the non-commuting graph of dihedral groups of order 2n, D2n, for all n ≥ 3.

Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups.

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph of a finite group is a graph where non-central elements of are its vertex set, while two different elements are edge connected when they do not commute in . In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of . We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

Twin g-noncommuting graph of a finite group

In this paper, we introduce the twin g-noncommuting graph of a finite group that is developed by combining the concepts of the g-noncommuting graph and the twin noncommuting graph of a finite group. The twin g-noncommuting graph of a finite group G, denoted by is constructed by considering the twin vertex set as one vertex and the adjacency of the two vertices are determined from their adjacency on the g-noncommuting graph. Furthermore, we choose dihedral group, whose representation of the twin g-noncommuting graph is determined. In addition, we determine the clique number of the twin g-noncommuting graph of dihedral group.