On minimal graphs, fixing sets and base size sets for hamiltonian groups

A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. Later, we determine the fixing sets and base size sets corresponding to finite hamiltonian groups. As a consequence, we obtain that if $H$ is a hamiltonian group, then the base size set of $H$ is equal to its fixing set.