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

Abstract

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.

Closeness Energy of Non-Commuting Graph for Dihedral Groups

“…Several authors have examined the energy of commuting and non-commuting graphs involving D 2n as the set of vertex. 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 .…”

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

“…Some researchers have published recent findings regarding the energy of 𝛤 𝐺 for 𝐷 2𝑛 , where 𝑛 ≥ 3. Degree exponent sum [13], maximum and minimum degree [14], degree subtraction [15], and neighbor degree sum [16] matrices were among the graph matrices they performed. As an extension of those investigations, the spectral radius and energy of 𝛤 𝐺 for 𝐷 2𝑛 corresponding with Seidel Laplacian and Seidel signless Laplacian matrices are discussed in this paper.…”

On the Non-Zero Divisor Graphs of Some Finite Commutative Rings

“…A ring is one of the fundamental algebraic structures consisting of a set with two binary operations which are addition and multiplication [5]. Several elementary results on the concept of rings and groups can be seen in Cohn [5], Auslander and Buchsbaum [2], Rotman [11], Bourbaki [4], Mudaber et al [6] and Romdhini et al [10].…”