Compressed Cayley graph of groups

Abstract

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