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

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