Malaysian Journal of Mathematical Sciences, March 2023, Vol. 17, No. 1
Neighbors Degree Sum Energy of Commuting and Non-Commuting Graphs for Dihedral Groups
Romdhini, M.U., Nawawi, A., and Chen, C.Y.
Corresponding Email: athirah@upm.edu.my
Received date: 11 August 2022
Accepted date: 11 January 2023
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 $2n$, $D_{2n}$, 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.
Keywords: commuting graph, non-commuting graph, dihedral group, neighbors degree sum matrix, the energy of a graph
