Maximum and Minimum Degree Energy of Commuting Graph for Dihedral Groups

Abstract

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by ΓG, has G\Z(G) as its vertices set with two distinct vertices vp and vq are adjacent if vp vq = vq vp. The degree of the vertex vp of ΓG, denoted by 𝑑𝑑𝑣𝑣𝑝𝑝 , is the number of vertices adjacent to vp. The maximum (or minimum) degree matrix of ΓG is a square matrix whose (p,q)-th entry is max{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 } (or min{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 }) whenever vp and vq are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of ΓG for dihedral groups of order 2n, D2n by using the absolute eigenvalues of the corresponding maximum degree matrices (MaxD(ΓG)) and minimum degree matrices (MinD(ΓG)). Here, the comparison of maximum and minimum degree energy of ΓG for D2n is discussed by considering odd and even n cases. The result shows that for each case, both energies are non-negative even integers and always equal.

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].…”

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

“…Clearly, 𝜌(𝛤 𝐺 ) is a non-negative real number and is the smallest disc radius that includes all the eigenvalues of 𝛤 𝐺 with the center at the origin of the complex plane [11]. There are a number of papers focusing on the spectral radius of other types of graphs, such as the spectral radius of power graphs on dihedral groups [5], the spectral radius of directed graphs [6] and the spectral radius of the interval-valued fuzzy graph [25] with regards to signless Laplacian matrix.…”

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

“…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.…”