On the symmetries of some classes of recursive circulant graphs

A recursive-circulant $G(n; d)$ is defined to be a circulant graph with $n$ vertices and jumps of powers of $d$. $G(n; d)$ is vertex-transitive, and has some strong hamiltonian properties. $G(n; d)$ has a recursive structure when $n = cd^m$, $1 leq c < d $ [10]. In this paper, we will find the automorphism group of some classes of recursive-circulant graphs. In particular, we will find that the automorphism group of $G(2^m; 4)$ is isomorphic with the group $D_{2 cdot 2^m}$, the dihedral group of order $2^{m+1}