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, 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.

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