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 ${{\,\textrm{Cay}\,}}(\mathbb {Z}_n,S)$.