Maximum and Minimum Degree Energy of Commuting Graph for Dihedral Groups

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by Gamma(G), has G\Z(G) as its vertices set with two distinct vertices v(p) and v(q) are adjacent if v(p) v(q) = v(q) v(p). The degree of the vertex v(p) of Gamma(G), denoted by d(nu p), is the number of vertices adjacent to v(p). The maximum (or minimum) degree matrix of Gamma(G) is a square matrix whose (p,q)-th entry is max{d(nu p), d(nu q)} (or min{d(nu p), d(nu q)}) whenever vp and vq are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of Gamma(G) for dihedral groups of order 2n, D-2n by using the absolute eigenvalues of the corresponding maximum degree matrices ( MaxD(Gamma(G))) and minimum degree matrices (MinD( Gamma(G))). Here, the comparison of maximum and minimum degree energy of Gamma(G) for D-2n is discussed by considering odd and even n cases. The result shows that for each case, both energies are non-negative even integers and always equal.