# graph product that commutes with automorphism, and semi direct

Is there a way to construct a "product" of graphs $G\rtimes H$ such that $Aut(G \rtimes H ) \simeq Aut(G) \rtimes Aut(H)$? A related topic is "Semidirect product" of graphs? but not quite the same.

• What action of the group $\text{Aut}(H)$ on the group $\text{Aut}(G)$ did you have in mind for defining their semidirect product? – Lee Mosher May 16 '15 at 13:42
• Any abstract action, the semi direct product of the graphs will depend on this action. – baharampuri May 17 '15 at 8:36
• For example if I want to find a graph with automorphism group isomorphic to $\mathbb{Z}_3 \rtimes \mathbb{Z}_9$ I don't have to follow Frucht's construction instead I can use this method. And contraction of graphs with automorphisms $\mathbb{Z}_3$ and $\mathbb{Z}_9$ is easier. – baharampuri May 18 '15 at 3:05
• Do you mean the group product? What type of product do you mean? – user1043065 May 18 '15 at 13:17
• Given an action of $Aut(H )$ on $Aut(G)$ I want a product to be defined on the two graphs $G$ and $H$ such that the automorphism group of this product is the semidirect product of the automorhpism groups of $G$ and $H$ under the action. – baharampuri May 18 '15 at 13:34