Let G be a group and S be a subset of G in which e /∈ S and S−1 ⊆ S. The Cayley graph of group G with respect to subset S, denoted by Cay(G, S), is an undirected simple graph whose vertices are all elements of G, and two vertices x and y are adjacent if and only if xy−1 ∈ S. If |S| = k, then Cay(G, S) is called a Cayley graph of valency k. The aim of this paper is to determine the structure of Cayley graph of dihedral groups D2n of order 2n when n = p or 2p2, where p is an odd prime number. The graph structures are based on circulant graphs with suitable jumps.

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