Degree Subtraction Energy of Commuting and Non-Commuting Graphs for Dihedral Groups

Let (Gamma) over bar (G) and Gamma(G) 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 (Gamma) over barG and is |G\Z(G)|, denoted by m. For the edge joining two distinct vertices v(p), v(q) is an element of G\Z(G) if and only if v(p)v(q) not equal v(q)v(p), on the other hand, whenever they commute in G, v(p) and v(q) are adjacent in (Gamma) over barG. The degree subtraction matrix (DSt) of Gamma(G) is denoted by DSt(Gamma(G) ), so that its (p, q)-entry is equal to d(vp) - d(vq), if v(p) not equal v(q), and zero if v(p) = v(q), where d(vp) is the degree of vp. For i = 1, 2,..., m, the maximum of |lambda(i) | as the DSt-spectral radius of and the sum of |lambda(i)| as DSt -energy of where lambda(i) are the eigenvalues of DSt(Gamma(G) ). These notations can be applied analogously to the degree subtraction matrix of the commuting graph, DSt( (Gamma) over bar G). Throughout this paper, we provide DSt-spectral radius and DSt-energy of and G for dihedral groups of order 2n, where n = 3. We then present the correlation of the energies and their spectral radius.