Let G be a group and let S be a subset of G \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ {e} with S−1 ⊆ S, where e is the identity element of G. The Cayley graph Cay(G, S) is a graph whose vertices are the elements of G and two distinct vertices g, h ∈ G are adjacent if and only if g−1h ∈ S. Let S ⊆ Z(G). Then the relation ∼ on G, given by a ∼ b if and onlyif Sa = Sb, is an equivalence relation. Let GE be the set of equivalence classes of ∼ on G and let [a] be the equivalence class of the element a in G. Then GE is a group with operation [a].[b]=[ab]. Also, let SE be the set of equivalence classes of the elements of S. The compressed Cayley graph of G is introduced as the Cayley graph Cay(GE , SE ), which is denoted by CayE (G, S). In this paper, we investigate some relations between Cay(G, S) and CayE (G, S). Also, we prove that Cay(G, S) is a CayE (G, S)-generalized join of certain empty graphs. Moreover, we describe the structure of the compressed Cayley graph of Zn by introducing a subset S such that CayE (Zn, S) and Cay(Zn, S) are not isomorphic, and we describe the Laplacian spectrum of Cay(Zn, S).

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