The hyper-star graph \(HS(n,k)\) is defined as follows: its vertex-set is the set of \(\{0, 1\}\)-sequences of length \(n\) with weight \(k\), where the weight of a sequence \(v\) is the number of \(1\)s in \(v\), and two vertices are adjacent if and only if one can be obtained from the other by exchanging the first symbol with a different symbol (\(1\) with \(0\), or \(0\) with \(1\)) in another position. In this paper, we will find the automorphism groups of regular hyper-star and folded hyper-star graphs. Then, we will show that only the graphs \(HS(4, 2)\) and \(FHS(4, 2)\) are Cayley graphs.