On minimal graphs for hamiltonian groups and their fixing set

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. The fixing number of a graph $\mathrm{\Gamma }$ is the minimum cardinality of a subset $S$ of $V(\mathrm{\Gamma })$ such that the stabilizer of $S$ is trivial. For a given finite group $G$, the fixing set is defined as the set comprising all possible fixing numbers of graphs having group $G$ as their automorphism groups. We determine the fixing sets corresponding to finite hamiltonian groups.