Compressed Cayley graph of groups

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