Degree-based energies of commuting graph for dihedral groups

Commuting graph for a finite group G, denoted by ΓG, with its set of vertices G\Z(G), where Z(G) is the center of G, is a graph with vp,vq ∈ G\Z(G), vp ≠ vq, are adjacent whenever vpvq = vq vp . In recent years, there has been significant research into the energy of graphs, particularly focusing on matrices associated with the degree of vertices. Therefore, motivated by that, our study elaborates on the energy of ΓG for dihedral groups of order 2n, D2n, concerning some graph matrices related to the degree of elements of D2n\Z(D2n) and examine the correlation between those energies. The matrices involved are known as geometric-arithmetic, symmetric division deg, degree exponent, inverse sum indeg and Sombor matrices. Based on these five matrices, it is found that the lowest graph energy is the geometric-arithmetic energy of ΓG whilst the highest is the degree exponent energy. Furthermore, the geometric-arithmetic, symmetric division deg, and degree exponent energies are always positive even integers. In contrast, the inverse sum indeg energy is a positive integer that can be either even or odd. Meanwhile, the Sombor energy is never an odd integer.

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