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All Journal Journal of the Indonesian Mathematical Society Science and Technology Indonesia
Nawawi, Athirah
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Biochemistry, Genetics & Molecular Biology Chemical Engineering, Chemistry & Bioengineering Environmental Science Materials Science & Nanotechnology Mathematics Physics

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Journal : Science and Technology Indonesia

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Relation Between the First Zagreb and Greatest Common Divisor Degree Energies of Commuting Graph for Dihedral Groups Romdhini, Mamika Ujianita; Nawawi, Athirah
Science and Technology Indonesia Vol. 10 No. 1 (2025): January
Publisher : Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26554/sti.2025.10.1.1-8

Abstract

The commuting graph for a finite group G, ΓG, has a set of vertices G \ Z(G), where Z(G) is the center of G, and vp,vq ∈ G \ Z(G) in which vp ≠ vq , are adjacent whenever vpvq = vqvp. The entries of the first Zagreb matrix (Z1) of ΓG are either the summation of the degrees of two adjacent vertices, or zero for non-adjacent vertices and also for the diagonal entries. Meanwhile, the entries of the greatest common divisor degree matrix (GCDD) of ΓG are the greatest common divisor of the degrees of two adjacent vertices and zero otherwise. The Z1-energy is determined by the sum of absolute eigenvalues of the corresponding Z1-matrix, whereas GCDD-energy is the sum of absolute eigenvalues of the GCDD-matrix. In this study, we find the spectral radius and the energies of ΓG for dihedral groups of order 2n, D2n, associated with Z1- and GCDD-matrices. It is found that Z1-energy is equal to twice GCDD-energy, whereas GCDD-energy is similar to maximum and minimum degree energies that were reported earlier in previous literature.
Co-Authors Al-Quran, Ashraf Al-Sharqi, Faisal Mamika Ujianita Romdhini, Mamika Ujianita