Relation between Randic and harmonic energies of commuting graph for dihedral groups

Consider a finite group G with center Z(G). This work examines the commuting graph ΓG, 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\Z(G), where two distinct vertices vp and vq are joined by an edge precisely when they commute, that is, whenever vpvq = vqvp . The number of vertices adjacent to vp is denoted as dvp, which is the degree of vp. The Randic and harmonic matrices of ΓG are defined as square matrices in which (p, q)−th entry are 1/√dvp·dvq and 2/dvp+dvq if vp and vq 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.

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