Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups
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 ϵ G \ Z(G) in which vp ≠ vq, are adjacent whenever vp vq = vq vp. The entries of the first Zagreb matrix (Z1) of ΓG are either the summation of the degrees of two adjacent vertices,...
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Magister Program of Material Sciences, Graduate School of Sriwijaya University
2025
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| Online Access: | http://psasir.upm.edu.my/id/eprint/121190/ http://psasir.upm.edu.my/id/eprint/121190/1/121190.pdf |
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| author | Romdhini, Mamika Ujianita Nawawi, Athirah |
| author_facet | Romdhini, Mamika Ujianita Nawawi, Athirah |
| author_sort | Romdhini, Mamika Ujianita |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | 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 ϵ G \ Z(G) in which vp ≠ vq, are adjacent whenever vp vq = vq vp. 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. |
| first_indexed | 2025-11-15T14:50:27Z |
| format | Article |
| id | upm-121190 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T14:50:27Z |
| publishDate | 2025 |
| publisher | Magister Program of Material Sciences, Graduate School of Sriwijaya University |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-1211902025-10-29T00:27:43Z http://psasir.upm.edu.my/id/eprint/121190/ Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups Romdhini, Mamika Ujianita Nawawi, Athirah 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 ϵ G \ Z(G) in which vp ≠ vq, are adjacent whenever vp vq = vq vp. 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. Magister Program of Material Sciences, Graduate School of Sriwijaya University 2025-01-01 Article PeerReviewed text en cc_by_4 http://psasir.upm.edu.my/id/eprint/121190/1/121190.pdf Romdhini, Mamika Ujianita and Nawawi, Athirah (2025) Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups. Science and Technology Indonesia, 10 (1). pp. 1-8. ISSN 2580-4405; eISSN: 2580-4391 https://sciencetechindonesia.com/index.php/jsti/article/view/1056 10.26554/sti.2025.10.1.1-8 |
| spellingShingle | Romdhini, Mamika Ujianita Nawawi, Athirah Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups |
| title | Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups |
| title_full | Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups |
| title_fullStr | Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups |
| title_full_unstemmed | Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups |
| title_short | Relation between the first Zagreb and Greatest Common Divisor degree energies of commuting graph for dihedral groups |
| title_sort | relation between the first zagreb and greatest common divisor degree energies of commuting graph for dihedral groups |
| url | http://psasir.upm.edu.my/id/eprint/121190/ http://psasir.upm.edu.my/id/eprint/121190/ http://psasir.upm.edu.my/id/eprint/121190/ http://psasir.upm.edu.my/id/eprint/121190/1/121190.pdf |