Let g be a finite group and s be a subset of g, where s does not include
the identity of g and is inverse closed. A cayley graph of a group g with respect to the
subset s is a graph, where its vertices are the elements of g and two vertices a and b
are connected if ab-1 is in the subset s. The energy of a cayley graph is the sum of all
absolute values of the eigenvalues of its adjacency matrix. In this paper, we consider a
specific subset s = {b, ab, . . . , An-1b} for dihedral groups of order 2n, where n 3 and find
the cayley graph with respect to the set. We also calculate the eigenvalues and compute
the energy of the respected cayley graphs. Finally, the generalization of the energy of the
respected cayley graphs is found.
Let G be a dihedral group and ??cl G its conjugacy class graph. The Laplacian energy of the graph, LE(??cl G) is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.