Connection between sum of Graphs and their automorphism groups

can we say something about the automorphism group of a graph $$G$$ that has the property: $$G \cong A + B$$ , if we know the automorphism groups of $$A$$ and $$B$$ respectively. The $$+$$ is the union $$\cup$$ of the $$A$$ and $$B$$ with the only addition that $$V(A) \cap V(B)= \emptyset$$.

Thank you in advance, any view on this would be helpful!

• Thank you for the answer my actual problem is that i have the situation: $\xoverline{K_{p,q}} \cong K_{p} + K_{q}$ where the graphs , are the complete graphs of p and q vertices respectively. Is it still the product? I mean i know that $Aut(K_{n})=S_{n}$ ..soo can I conclude to the fact that $Aut(\xoverline{K_{p,q}}=S_{p} \times S{q}$? – Someone86 Mar 14 at 13:22
• Absolutely! Any pair of permutations of $p,q$ still corresponds to an element of the automorphism group. Don't forget that this is only if $p\neq q$, because otherwise we get things where the points can be mapped to the other subgraph. – Michael Gintz Mar 14 at 16:06