The Energy of Cayley Graphs for a Generating Subset of the Dihedral Groups

Abstract

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, . . . , a^(n−1)b} for dihedral group of order 2n, where n is greater or equal to 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.

Power Cayley Graphs of Dihedral Groups with Certain Order

“…⊆ 𝑆 , denoted as Cay(G, S), is a graph with elements of 𝐺 as its vertices and two vertices 𝑥 and 𝑦 are adjacent if there exist 𝑠 ∈ 𝑆 such that 𝑥 = 𝑦𝑠 or 𝑦 = 𝑥𝑠. Recent results related to the Cayley graph can be referred in the previous studies [2][3][4][5][6].…”