On the symmetries of some classes of recursive circulant graphs

Abstract

A recursive-circulant $G(n; d)$ is defined to be a‎ ‎circulant graph with $n$ vertices and jumps of powers of $d$‎. <br />‎$G(n; d)$ is vertex-transitive‎, ‎and has some strong hamiltonian‎ ‎properties‎. ‎$G(n;d)$ has a recursive structure when $n = cd^m$‎, ‎$1 leq c < d $ [<em>Theoret‎. ‎Comput‎. ‎Sci.</em> <strong>244</strong> (2000) 35-62]‎. ‎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 $2^{m+1}$‎.