The Structure of the Generalized Cayley Graph Cayₘ (〖 D〗₂n, S) of Valency Three

Suppose that G is a finite group and S is a non-empty subset of G such that e∉S and S^ (-1) ⊆S. Suppose that Cay (G, S) is the Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy^ (-1) ∈S. In this paper, we introduce the generalized Cayley graph denoted by Cayₘ (G, S) that is a graph with vertex set consists of all column matrices Xₘ which all components are in G and two vertices Xₘ and Yₘ are adjacent if and only if Xₘ (Yₘ) ^ (-1) ᵗ∈M (S), where 〖Yₘ〗^ (-1) is a column matrix that each entry is the inverse of similar entry of Yₘ and M (S) is m×m matrix with all entries in S, Y^ (-1) ᵗ is the transpose of Y^ (-1) and m≥1. We aim to determine the structure of Cayₘ (G, S) when G is the dihedral group of order 2n and | S |= 3 for every m≥2, n≥3.