CLOSENESS LAPLACIAN AND CLOSENESS SIGNLESS LAPLACIAN ENERGIES OF NON-COMMUTING GRAPH FOR DIHEDRAL GROUPS

Abstract: This research investigates the relationship between algebra and graph theory, specifically how algebra facilitates graph theory. Associating matrices with graphs introduces the concept of graph energies. A new energy formula of non-commuting graphs for dihedral groups using closeness Laplacian and closeness signless Laplacian matrices is investigated in this paper. It is found that both energies are always equivalent and are categorized as hyperenergetic.

Keywords: energy of a graph, non-commuting graph, dihedral group, closeness Laplacian matrix, closeness signless Laplacian matrix.

MSC: 05C25, 15A18

DOI       10.56082/annalsarscimath.2026.2.145

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^†mamika@unram.ac.id, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia
^‡abdurahim@staff.unram.ac.id, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia
§a.ellena.saufika@staff.unram.ac.id, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia
^¶athirah@upm.edu.my, Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
^||faisal.ghazi@uoanbar.edu.iq, Department of Mathematics, Faculty of Education for Pure Sciences, University Of Anbar, Ramadi, Anbar, Iraq; College of Engineering, National University of Science and Technology, Dhi Qar, Iraq; Department of Mathematics, College of Education, Al-Ayen Iraqi University, An Nasiriyah, Iraq
^**ifanhasnandani@gmail.com, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia

PUBLISHED in

Annals Academy of Romanian Scientists Series on Mathematics and Its Application,

Volume 18 no 2, 2026

ISSN ONLINE 2066 – 6594
ISSN PRINT 2066 – 5997