Relation Between Randic and Harmonic Energies of Commuting Graph for Dihedral Groups - Romdhini, Mamika Ujianita; Nawawi, Athirah (Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya, 2026)
Consider a finite group G with center Z(G). This work examines the commuting graph Γ_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 GZ(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 Γ_G are defined as square matrices in which (p,q)-th entry are 1d_v_p · d_v_q and 2d_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.