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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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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.
On Spectrum and Energy of Non-Commuting Graph for Group U_{6n} Romdhini, Mamika Ujianita; Nawawi, Athirah; Al-Sharqi, Faisal; Al-Quran, Ashraf
Journal of the Indonesian Mathematical Society Vol. 31 No. 4 (2025): DECEMBER
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.v31i4.1928

Abstract

Let $G$ be a group and $Z(G)$ be the center of $G$. In this paper, we discuss a specific type of graph known as the non-commuting graph, denoted by $\Gamma_G$, whose vertex set contains all group elements excluding central elements, $G\backslash Z(G)$. This graph has to satisfy a condition in which $v_p,v_q \in G\backslash Z(G)$ where $v_p \neq v_q$, are adjacent whenever $v_p v_q\neq v_q v_p$. This paper presents the spectrum and energy of the non-commuting graph for $U_{6n}$, $\Gamma_{U_{6n}}$ associated with the adjacency, degree sum, and degree sum adjacency matrices and their energy relationship.
Relation Between Randic and Harmonic Energies of Commuting Graph for Dihedral Groups Romdhini, Mamika Ujianita; Nawawi, Athirah
Science and Technology Indonesia Vol. 11 No. 2 (2026): April
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.2026.11.2.481-488

Abstract

Consider a finite group G with center Z(G). This work examines the commuting graph $\Gamma_G$, a graph constructed from a group $G$ whose vertices correspond precisely to the noncentral elements of the group, that is, all elements in $G$ except those belonging to its center $Z(G)$. The graph is defined on the vertex set $G\backslash Z(G)$, where two distinct vertices vp and vq are joined by an edge precisely when they commute, that is, whenever vp vq=vq vp. The number of vertices adjacent to vp is denoted as dvp, which is the degree of vp. The Randic and harmonic matrices of $\Gamma_G$ are defined as square matrices in which $(p,q)-$th entry are $\frac{1}{\sqrt{d_{v_p} \cdot d_{v_q}} }$ and $\frac{2}{d_{v_p}+d_{v_q} }$ if $v_p$ and $v_q$ are adjacents, respectively; otherwise, it is zero. Randic energy is the sum of the absolute eigenvalues of the Randic matrix whereas harmonic energy is the sum of the absolute eigenvalues of the harmonic matrix. In this paper, we compare the Randic and harmonic energies of the commuting graph for non-abelian dihedral group of order 2n, D2n.
Co-Authors Al-Quran, Ashraf Al-Sharqi, Faisal Mamika Ujianita Romdhini, Mamika Ujianita