Relation Between Randic and Harmonic Energies of Commuting Graph for Dihedral Groups

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 v_p and vq are joined by an edge precisely when they commute, that is, whenever v_p v_q=v_q v_p. The number of vertices adjacent to v_p is denoted as d_vp, which is the degree of v_p. 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, D_2n.