Degree subtraction energy of commuting and non-commuting graphs for dihedral groups

Let ¯ ΓG and ΓG be the commuting and non-commuting graphs on a finite group G, respectively, having G\Z(G) as the vertex set, where Z(G) is the center of G. The order of ¯ ΓG and ΓG is |G\Z(G)|, denoted by m. For ΓG, the edge joining two distinct vertices vp,vq ∈ G\Z(G) if and only if vpvq= vqvp, on the other hand, whenever they commute in G, vp and vq are adjacent in ¯ ΓG. The degree subtraction matrix (DSt) of ΓG is denoted by DSt(ΓG), so that its (p,q)−entry is equal to dvp − dvq , if vp= vq, and zero if vp = vq, where dvp is the degree of vp. For i =1,2,...,m, the maximum of |λi| as the DSt−spectral radius of ΓG and the sum of |λi| as DSt−energy of ΓG, where λi are the eigenvalues of DSt(ΓG). These notations can be applied analogously to the degree subtraction matrix of the commuting graph, DSt(¯ ΓG). Throughout this paper, we provide DSt−spectral radius and DSt−energy of ΓG and ¯ ΓG for dihedral groups of order 2n, where n ≥ 3. We then present the correlation of the energies and their spectral radius.

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