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.

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Received date: 11 August 2022
Accepted date: 11 January 2023

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



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