The automorphism groups of minimal infinite circulant digraphs (Q1357267)

Given a group \(G\), a directed graph \(X\) is called a digraph regular representation (DRR) of \(G\) if Aut\((X)\simeq G\) (as abstract groups) and the action of Aut\((X)\) on \(V(X)\) is that of a regular permutation group. It is well known that in this case \(X\) must be a Cayley digraph \(X(G,S)\) where \(S\subset G\) and \(\langle S\rangle=G\). Let \(Z\) denote the infinite cyclic group. Under the assumption that \(X=X(Z,S)\) is strongly connected, the following results are proved: if \(S\) is a minimal generating set (with respect to \(\subset\)) of \(Z\), then \(X\) is a DRR of \(Z\) with Aut\((X)\) consisting of all translations \(x\mapsto x+a\) for \(a\in Z\); if \(|S|=2\), then \(X\) is a DRR if and only if \(S\neq\{-1,1\}\); if \(S\) is a minimal generating set of \(Z\), then \(X(Z,S)\cong X(Z,T)\) if and only if \(S=\pm T\). It is conjectured that, if \(X(Z,S)\) is strongly connected and \(S\neq-S\), then \(X(Z,S)\) is a DRR, while if \(S=-S\), then Aut\((X)\simeq D_\infty\).