Infinite line as a Cayley graph
= Infinite line as a Cayley graph
The undirected infinite line is both the <Cayley graph> of the <infinite cyclic group> $\mathbb Z$ with generating set $\{1\}$ and the Cayley graph of the <infinite dihedral group> $C_2*C_2$ with its two involutory generators. Thus an unlabelled Cayley graph need not determine its group.