Degree Sum Energy of Non-Commuting Graph for Dihedral Groups

Abstract

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.

Spectral Properties of Power Graph of Dihedral Groups

“…Several authors have discussed the graphs that are defined on dihedral groups. They worked on the spectral problem of the commuting and non-commuting graphs, which can be seen in [9][10][11][12][13], Accordingly, Romdhini et al [8] investigated signless Laplacian spectral of interval-valued fuzzy graphs. Moreover, an analysis of the relationship between graphs and unitary commutative rings' prime spectrum is presented by [1].…”

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

“…In the present work, we focus on the non-abelian dihedral groups of order 2𝑛, 𝑛 ≥ 3, denoted by 𝐷 2𝑛 = 〈𝑎, 𝑏 ∶ 𝑎 𝑛 = 𝑏 2 = 𝑒, 𝑏𝑎𝑏 = 𝑎 −1 〉 and its elements can be written as 𝑎 𝑖 and 𝑎 𝑖 𝑏 [3]. Many researchers are currently interested in studying the characteristic polynomial of graphs, for instance, the signless Laplacian polynomial for simple graphs [13] and the characteristic polynomial based on the Sombor matrix [14]. Moreover, the Laplacian spectrum of coprime order graph for finite abelian 𝑝-group has been presented in [21].…”

“…Moreover, the Laplacian spectrum of coprime order graph for finite abelian 𝑝-group has been presented in [21]. Meanwhile, a discussion of characteristic polynomials based on degree-based matrices applied to commuting and non-commuting graphs for 𝐷 2𝑛 can be seen in [15,16,17,18,19]. These results motivate us to investigate the characteristic polynomial of the power graph of the dihedral graph associated with degree-based matrices.…”

Seidel Laplacian and Seidel Signless Laplacian Energies of Commuting Graph for Dihedral Groups

“…The spectral radius of 𝛤 𝐺 is denoted by the formula 𝜌(𝛤 𝐺 ) = 𝑚𝑎𝑥{|𝜆|: 𝜆 ∈ 𝑆𝑝𝑒𝑐(𝛤 𝐺 )} [7]. Several scholarly articles examine the spectral radius and spectrum of alternative graph types, including the non-commuting graph [17] in relation to the Sombor matrix, the coprime graph [18], and the cubic power graph [12].…”