The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

Abstract

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

The Structure of the Generalized Cayley Graph Cay_m (〖 D〗_2n,S) of Valency Three

“…Theorem 1.5. [9] Assume that S = {x, x −1 , y}⊆D2n such that x≠x −1 and y 2 = e and o(x)=m. Then Cay(D2n, S) = n m (K2Cn).…”

“…Theorem 1.6. [9] Let S = {x, y, z}}⊆D2n , with x 2 = y 2 = z 2 = e. Then Cay(D2n, S) = K2Cn. To determine the graph structure of Cay(D2n, S) for |S| = 3, it is enough to consider on of the above two cases.…”

“…𝐺 ∘ 𝐻 ≠ 𝐻 ∘ 𝐺. Throughout this paper, we always assume that group G is finite, S is a nonempty subset of G, e ∉ S, S −1 = S and S is a genetaing set for G. Moreover, all of the notations and terminologies about graphs are standard and can be found in [9].…”