On the non-commuting graph of dihedral group

Khasraw, Sanhan Muhammad Salih; Ali, Ivan; Haji, Rashad

doi:10.5614/ejgta.2020.8.2.3

Cited by 7 publications

(6 citation statements)

References 10 publications

(11 reference statements)

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“…10.11113/mjfas.v20n1.3252 Theorem 2.1. [12] Let 𝛤 𝐺 be the non-commuting graph for 𝐺, where 𝐺 = 𝐺 1 ∪ 𝐺 2 . Then 1. the degree of 𝑎 𝑖 on 𝛤 𝐺 is 𝑑 𝑎 𝑖 = 𝑛, and 2. the degree of 𝑎 𝑖 𝑏 on 𝛤 𝐺 is 𝑑 𝑎 𝑖 𝑏 = { 2(𝑛 − 1), if 𝑛 is odd 2(𝑛 − 2), if 𝑛 is even.…”

Section: Preliminariesmentioning

confidence: 99%

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

Romdhini,

Nawawi

2024

Mal. J. Fund. Appl. Sci.

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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.

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Section: Preliminariesmentioning

confidence: 99%

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

Romdhini,

Nawawi

2024

Mal. J. Fund. Appl. Sci.

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“…Theorem 2.1: (Khasraw et al, 2020) The following lemma helps us to compute the characteristic polynomial of the non-commuting graph of 𝑫 𝟐𝒏 .…”

Section: Preliminariesmentioning

confidence: 99%

“…This refutes the conjecture by Gutman et al in 2008, stating that the adjacency energy of any graph is smaller than or equal to its Laplacian energy, which holds for all graphs. However, readers can also see different perspectives of this particular graph where the discussion on the detour index, eccentric connectivity, total eccentricity polynomials, and mean distance of the non-commuting graph for the dihedral group by Khasraw et al (2020).…”

Section: Introductionmentioning

confidence: 99%

Degree Sum Energy of Non-Commuting Graph for Dihedral Groups

Romdhini

Nawawi²

2022

MJS

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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.

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“…The study of graphical representations of algebraic structures, especially groups, has been an energizing and fascinating research area originating from the pioneering paper by Arthur Cayley [26] with many recent results (cf. [11,14,40,54,79]). In particular, graphs associated to groups and other algebraic constructions have valuable applications (cf.…”

Section: Introductionmentioning

confidence: 99%