Degree Exponent Sum Energy of Commuting Graph for Dihedral Groups

Abstract

For a finite group G and a nonempty subset X of G, we construct a graph with a set of vertex X such that any pair of distinct vertices of X are adjacent if they are commuting elements in G. This graph is known as the commuting graph of G on X, denoted by ΓG [X]. The degree exponent sum (DES) matrix of a graph is a square matrix whose (p,q)-th entry is is dvp dvq + dvqdvp whenever p is different from q, otherwise, it is zero, where dvp (or dvq ) is the degree of the vertex vp (or vertex, vq) of a graph. This study presents results for the DES energy of commuting graph for dihedral groups of order 2n, using the absolute eigenvalues of its DES matrix.

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

“…The centralizer of the element 𝑎 𝑖 in 𝐷 2𝑛 is 𝐶 𝐷 2𝑛 (𝑎 𝑖 ) = { 𝑎 𝑗 : 1 ≤ 𝑗 ≤ 𝑛 } and for the element 𝑎 𝑖 𝑏 is either 𝐶 𝐷 2𝑛 (𝑎 𝑖 𝑏) = {𝑒, 𝑎 𝑖 𝑏}, if 𝑛 is odd or 𝐶 𝐷 2𝑛 (𝑎 𝑖 𝑏) = {𝑒, 𝑎 𝑛 2 , 𝑎 𝑖 𝑏, 𝑎 𝑛 2 +𝑖 𝑏}, if 𝑛 is even. Some recent results in the energy of commuting and non-commuting graphs for 𝐷 2𝑛 , for 𝑛 ≥ 3, denoted by 𝐷 2𝑛 have been reported in [16,24]. They worked on adjacency and degree exponent sum matrices.…”

Closeness Energy of Non-Commuting Graph for Dihedral Groups

“…By considering the eigenvalues of the degree sum and degree subtraction matrices, Romdhini and Nawawi [17,19] and Romdhini et al [22] formulated the energy. In [18,21], the sum of the degree exponent and the maximum and minimum degree energies were presented for D 2n . Therefore, the purpose of this paper is to formulate the energy based on the closeness matrix for Γ G on D 2n .…”