A graph $\Gamma$ is called prime with respect to the Cartesian product if $\Gamma=\Gamma_1\square\Gamma_2$ implies that $\Gamma_1=K_1$ or $\Gamma_2=K_1$, where $\square$ denote the Cartesian product. Is there any classification of connected and vertex-transitive prime graphs with respect to Cartesian product? Is there any information about the automorphism group of such graphs?
There almost certainly isn't a meaningful classification of vertex-transitive prime graphs. In fact, it's quite likely that almost all vertex-transitive graphs are prime.
It's all in Hammock, Imrich and Klavzar "Handbook of Product Graphs". The rough summary is that everything works nicely, and you do not need transitivity. The automorphism group will be the direct product of wreath products.