On the Spectral Characterization of Prime Coprime Graphs Associated with Dihedral Groups

Mamika Ujianita Romdhini (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram, 83125, Indonesia)
Abdurahim (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram, 83125, Indonesia)
Adem Kilicman (Department of Mathematical Sciences, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Salangor, 40450, Malaysia)
Dragan Pamucar (Department of Engineering and Natural Sciences, Istinye University, Istanbul, 34396, Turkiye)
Faisal Al-Sharqi (Department of Mathematics, College of Education for Pure Sciences, University Of Anbar, Ramadi, Anbar, 55431, Iraq)
Ifan Hasnan Dani (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram, 83125, Indonesia)
Marliadi Susanto (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram, 83125, Indonesia)

Science and Technology Indonesia
Vol. 11 No. 4 (2026): October

Spectral graph theory and group theory help reveal how the structure of graphs shapes their mathematical properties. This research aims to analyze the spectral descriptors of the prime coprime graph of a dihedral group associated with the Laplacian, normalized Laplacian, and Seidel matrices. We also establish a method for determining the eigenvalues of these matrices. The results show that the Laplacian, signless Laplacian, and normalized Laplacian energies of the obtained graph are always even integers, whereas the adjacency and Seidel energies are never odd integers. These findings are consistent with previous studies and further strengthen the role of spectral descriptors in understanding algebraicgraph structures.